Pentagonal number theorem information

In mathematics, Euler's pentagonal number theorem relates the product and series representations of the Euler function. It states that ∏ n = 1 ∞ ( 1 −...

involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern...

is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. p ( n ) =...

(triangular numbers) 17 = 16 + 1 (square numbers) 17 = 12 + 5 (pentagonal numbers). The theorem is named after Pierre de Fermat, who stated it, in 1638, without...

(number theory) Integer partition Bell numbers Landau's function Pentagonal number theorem Bell series Lambert series Twin prime Brun's constant Cousin prime...

the partition function. The Euler identity, also known as the Pentagonal number theorem, is ϕ ( q ) = ∑ n = − ∞ ∞ ( − 1 ) n q ( 3 n 2 − n ) / 2 . {\displaystyle...

pentagonal numbers (OEIS: A001318, starting at offset 1). Indeed, Euler proved this by logarithmic differentiation of the identity in his pentagonal number...

four squares. Euler's identity may also refer to the pentagonal number theorem. Euler's number, e = 2.71828..., the base of the natural logarithm Euler's...

formula Ferrers graph Glaisher's theorem Landau's function Partition function (number theory) Pentagonal number theorem Plane partition Quotition and partition...

}^{\infty }(-1)^{n}q^{\frac {3n^{2}-n}{2}}.} Note that by using Euler Pentagonal number theorem for I ( τ ) > 0 {\displaystyle {\mathfrak {I}}(\tau )>0} , the...

(physics) Pentagonal number theorem (number theory) Perfect graph theorem (graph theory) Perlis theorem (graph theory) Perpendicular axis theorem (physics)...

Iterated binary operation Infinite expression Infinite series Pentagonal number theorem Jeffreys, Harold; Jeffreys, Bertha Swirles (1999). Methods of...

proof of a classical result on the number of certain integer partitions. Bijective proofs of the pentagonal number theorem. Bijective proofs of the formula...

corresponding affine Kac–Moody algebra. Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi Triple Product...

pentagonal number and the (3k − 1)th pentatope number is always the ( 3 k 2 + k 2 ) {\displaystyle \left({\tfrac {3k^{2}+k}{2}}\right)} th pentagonal...

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle...

on 2013-05-29. Retrieved 2010-05-13. Weisstein, Eric W. "Pentagonal Square Triangular Number". MathWorld. The Penguin Dictionary of Curious and Interesting...

summandization of the now mentioned Pochhammer product is described by the Pentagonal number theorem in this way: ( x ; x ) ∞ = 1 + ∑ n = 1 ∞ [ − x Fn ( 2 n − 1 )...

echinoderms with a pentagonal shape. A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron. The faces are true regular pentagons. A pyritohedral...

{3j+1}{2}}\right\rceil } denote the sequence of interleaved pentagonal numbers, i.e., so that the pentagonal number theorem is expanded in the form of ( q ; q ) ∞ = ∑...

smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime...

contains a statement of Euler's formula and a statement of the pentagonal number theorem, which he had discovered earlier and would publish a proof for...

numbers, integer partitions, and the generating functions and pentagonal number theorem that connect these two concepts. In chapter 6, Reid brings in...

certain non-reduced affine root system. It is related to Euler's pentagonal number theorem. ∏ n ≥ 1 ( 1 − s n ) ( 1 − s n t ) ( 1 − s n − 1 t − 1 ) ( 1 −...

first equality. The second equality can be proved by using the pentagonal number theorem. Berndt, Bruce C. (1998). Ramanujan's Notebooks Part V. Springer...

{1}{2}}k(3k-1))+P(n-{\frac {1}{2}}k(3k+1))]} derived from Euler's pentagonal number theorem. Written as a dfn:: §16  pn ← {1≥⍵:0≤⍵ ⋄ -⌿+⌿∇¨rec ⍵} rec ← {⍵...