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A322211 a(n) = coefficient of x^n*y^n in Product_{n>=1} 1/(1 - (x^n + y^n)). 6
1, 2, 10, 38, 158, 602, 2382, 9142, 35492, 136936, 530404, 2053848, 7972272, 30977742, 120576112, 469915012, 1833813534, 7164469910, 28021000340, 109699469798, 429850240742, 1685728936622, 6615913739206, 25983523253950, 102115250446680, 401557335718522, 1579978592844064, 6219928993470190, 24498287876663618, 96535916978924934, 380568644820360668 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of subsets of partitions of 2n that have sum n. Olivier Gérard, May 07 2020

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..500 (previous b-file of terms 0..50 supplied by Vaclav Kotesovec).

FORMULA

Main diagonal of square table A322210.

a(n) ~ c * 4^n / sqrt(Pi*n), where c = 1 / A048651 = 1 / Product_{k>=1} (1 - 1/2^k) = 3.46274661945506361153795734292443116454075790290443839... - Vaclav Kotesovec, Dec 23 2018

EXAMPLE

G.f.: A(x) = 1 + 2*x + 10*x^2 + 38*x^3 + 158*x^4 + 602*x^5 + 2382*x^6 + 9142*x^7 + 35492*x^8 + 136936*x^9 + 530404*x^10 + 2053848*x^11 + 7972272*x^12 + ...

RELATED SERIES.

The product P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)) begins

P(x,y) = 1 + (x + y) + (2*x^2 + 2*x*y + 2*y^2) + (3*x^3 + 4*x^2*y + 4*x*y^2 + 3*y^3) + (5*x^4 + 7*x^3*y + 10*x^2*y^2 + 7*x*y^3 + 5*y^4) + (7*x^5 + 12*x^4*y + 18*x^3*y^2 + 18*x^2*y^3 + 12*x*y^4 + 7*y^5) + (11*x^6 + 19*x^5*y + 34*x^4*y^2 + 38*x^3*y^3 + 34*x^2*y^4 + 19*x*y^5 + 11*y^6) + (15*x^7 + 30*x^6*y + 56*x^5*y^2 + 74*x^4*y^3 + 74*x^3*y^4 + 56*x^2*y^5 + 30*x*y^6 + 15*y^7) + (22*x^8 + 45*x^7*y + 94*x^6*y^2 + 133*x^5*y^3 + 158*x^4*y^4 + 133*x^3*y^5 + 94*x^2*y^6 + 45*x*y^7 + 22*y^8) + ...

in which this sequence equals the coefficients of x^n*y^n for n >= 0.

The logarithm of the g.f. begins

log( A(x) ) = 2*x + 16*x^2/2 + 62*x^3/3 + 272*x^4/4 + 922*x^5/5 + 3640*x^6/6 + 12966*x^7/7 + 49872*x^8/8 + 190340*x^9/9 + 745316*x^10/10 + 2928136*x^11/11 + 11602184*x^12/12 + ...

MATHEMATICA

nmax = 20; s = Series[Product[1/(1 - (x^k + y^k)), {k, 1, nmax}], {x, 0, nmax}, {y, 0, nmax}]; Flatten[{1, Table[Coefficient[s, x^n*y^n], {n, 1, nmax}]}] (* Vaclav Kotesovec, Dec 04 2018 *)

PROG

(PARI)

{P = 1/prod(n=1, 61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) ); }

{a(n) = polcoeff( polcoeff( P, n, x), n, y)}

for(n=0, 35, print1( a(n), ", ") )

CROSSREFS

Cf. A258471, A322210, A322213, A322198.

Sequence in context: A166898 A143960 A122117 * A120949 A186097 A295521

Adjacent sequences:  A322208 A322209 A322210 * A322212 A322213 A322214

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 30 2018

STATUS

approved

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Last modified February 25 08:33 EST 2021. Contains 341606 sequences. (Running on oeis4.)