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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A322211 a(n) = coefficient of x^n*y^n in Product_{n>=1} 1/(1 - (x^n + y^n)).

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2018

Keywords

Comments

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

Examples

			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 + ...

Links

Crossrefs

Cf. A108796, A258471, A322210, A322213, A322198.

Programs

  • 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 *)
  • 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),", ") )

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

A322187 a(n) is the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) ), for n >= 1.

Original entry on oeis.org

1, 3, 13, 35, 131, 471, 1723, 6435, 24349, 92393, 352727, 1352183, 5200313, 20058321, 77559203, 300540195, 1166803127, 4537569063, 17672631919, 68923264585, 269128942459, 1052049481893, 4116715363823, 16123801860855, 63205303219531, 247959266474091, 973469712897103, 3824345300380465, 15033633249770549, 59132290782710201, 232714176627630575, 916312070471295267, 3609714217009191161, 14226520737620288421
Offset: 1

Views

Author

Paul D. Hanna, Dec 07 2018

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 35*x^4/4 + 131*x^5/5 + 471*x^6/6 + 1723*x^7/7 + 6435*x^8/8 + 24349*x^9/9 + 92393*x^10/10 + 352727*x^11/11 + 1352183*x^12/12 + ...
RELATED SERIES.
exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 45*x^5 + 140*x^6 + 448*x^7 + 1483*x^8 + 5027*x^9 + 17311*x^10 + 60469*x^11 + 213678*x^12 + ... + A322188(n)*x^n + ...
The table of coefficients of x^n*y^k/(n+k) in
log( Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) ) = (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2)/2 + (4*x^3 + 3*x^2*y + 3*x*y^2 + 4*y^3)/3 + (1*x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + 1*y^4)/4 + (6*x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + 6*y^5)/5 + (4*x^6 + 6*x^5*y + 15*x^4*y^2 + 26*x^3*y^3 + 15*x^2*y^4 + 6*x*y^5 + 4*y^6)/6 + (8*x^7 + 7*x^6*y + 21*x^5*y^2 + 35*x^4*y^3 + 35*x^3*y^4 + 21*x^2*y^5 + 7*x*y^6 + 8*y^7)/7 + (1*x^8 + 8*x^7*y + 28*x^6*y^2 + 56*x^5*y^3 + 70*x^4*y^4 + 56*x^3*y^5 + 28*x^2*y^6 + 8*x*y^7 + 1*y^8)/8 + ...
begins
n=0: [0, 1, 1, 4, 1, 6, 4, 8, 1, 13, 6, ..., A000593(k), ...];
n=1: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...];
n=2: [1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...];
n=3: [4, 4, 10, 26, 35, 56, 93, 120, 165, 232, ...];
n=4: [1, 5, 15, 35, 70, 126, 210, 330, 495, 715, ...];
n=5: [6, 6, 21, 56, 126, 262, 462, 792, 1287, 2002, ...];
n=6: [4, 7, 28, 93, 210, 462, 942, 1716, 3003, 5035, ...];
n=7: [8, 8, 36, 120, 330, 792, 1716, 3446, 6435, 11440, ...];
n=8: [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, ...];
n=9: [13, 10, 55, 232, 715, 2002, 5035, 11440, 24310, 48698, ...];
n=10: [6, 11, 66, 286, 1001, 3018, 8008, 19448, 43758, 92378, ...]; ...
in which the diagonal of coefficients of x^n*y^n/(2*n) equals
[0, 2, 6, 26, 70, 262, 942, 3446, 12870, 48698, ..., 2*a(n), ...],
which is twice this sequence.
The related infinite product may be written as the following series expansion
Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) = 1/( (1 - x - y) * (1 - x^3 - y^3) * (1 - x^5 - y^5) * (1 - x^7 - y^7) * (1 - x^9 - y^9) * (1 - x^11 - y^11) * ...) = 1 + (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2) + (2*x^3 + 3*x^2*y + 3*x*y^2 + 2*y^3) + (2*x^4 + 5*x^3*y + 6*x^2*y^2 + 5*x*y^3 + 2*y^4) + (3*x^5 + 7*x^4*y + 11*x^3*y^2 + 11*x^2*y^3 + 7*x*y^4 + 3*y^5) + (4*x^6 + 10*x^5*y + 18*x^4*y^2 + 24*x^3*y^3 + 18*x^2*y^4 + 10*x*y^5 + 4*y^6) + (5*x^7 + 14*x^6*y + 28*x^5*y^2 + 42*x^4*y^3 + 42*x^3*y^4 + 28*x^2*y^5 + 14*x*y^6 + 5*y^7) + (6*x^8 + 19*x^7*y + 42*x^6*y^2 + 71*x^5*y^3 + 84*x^4*y^4 + 71*x^3*y^5 + 42*x^2*y^6 + 19*x*y^7 + 6*y^8) + ...

Links

Crossrefs

Cf. A322188, A322198, A322203, A000593.

Programs

  • PARI
    N=35;
{L = sum(n=1, N+1, -log(1 - x^(2*n-1) - y^(2*n-1) +x*O(x^N) +y*O(y^N)) ); }
{a(n) = polcoeff( n*polcoeff( L, n, x), n, y)}
for(n=1, N, print1( a(n), ", ") )

Formula

a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 18 2019

A322188 G.f.: exp( Sum_{n>=1} A322187(n)*x^n/n ), where A322187(n) is the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) ).

Original entry on oeis.org

1, 1, 2, 6, 15, 45, 140, 448, 1483, 5027, 17311, 60469, 213678, 762284, 2741864, 9932346, 36202666, 132677658, 488605698, 1807176452, 6710206574, 25003642942, 93468147306, 350425771854, 1317330452697, 4964398631867, 18751217069083, 70975750129731, 269180061675328, 1022750160098864, 3892577330120307, 14838784128136803, 56651259287153670, 216586672901518164, 829142137823283601, 3178107527615273349
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 45*x^5 + 140*x^6 + 448*x^7 + 1483*x^8 + 5027*x^9 + 17311*x^10 + 60469*x^11 + 213678*x^12 + ...
such that
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 35*x^4/4 + 131*x^5/5 + 471*x^6/6 + 1723*x^7/7 + 6435*x^8/8 + 24349*x^9/9 + 92393*x^10/10 + 352727*x^11/11 + 1352183*x^12/12 + ... + A322187(n)*x^n/n + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 46*x^4 + 144*x^5 + 466*x^6 + 1536*x^7 + 5187*x^8 + 17842*x^9 + 62209*x^10 + 219504*x^11 + 782272*x^12 + ...

Links

Crossrefs

Cf. A322187, A322198, A322204.

Programs

  • PARI
    N=35;
{L = sum(n=1, N+1, -log(1 - x^(2*n-1) - y^(2*n-1) +x*O(x^N) +y*O(y^N)) ); }
{A322187(n) = polcoeff( n*polcoeff( L, n, x), n, y)}
{a(n) = polcoeff( exp( sum(m=1, n, A322187(m)*x^m/m ) +x*O(x^n) ), n) }
for(n=0, N, print1( a(n), ", ") )

Formula

a(n) ~ c * 4^n / n^(3/2), where c = 0.57389010009720382786456367148681469430628117317... - Vaclav Kotesovec, Jun 18 2019
Showing 1-3 of 3 results.

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