cp's OEIS Frontend

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 1-5 of 5 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

A322214 a(n) = coefficient of x^n*y^n in Product_{n>=1} (1 - (x^n + y^n))^3.

Original entry on oeis.org

1, 6, -12, -6, 6, -12, -12, 96, 24, -134, -192, 114, 736, 282, -792, -1532, -270, 1932, 2004, -96, -3654, -6910, -5532, 4836, 21500, 23454, 11850, -8216, -43998, -57744, -34424, 16716, 73506, 105500, 87432, -24474, -230028, -331626, -257616, -163250, 316434, 852450, 1130284, 1175748, 361110, -652820, -1956330, -2964180, -2922288, -1965174, 187806, 3863602, 6585672, 6996900, 6199180, 366768, -7228866, -14682152, -21063366, -19602108, -10562926, 6959976, 30061386, 50110338, 66753126, 68131632, 37666392
Offset: 0

Views

Author

Paul D. Hanna, Dec 04 2018

Keywords

Comments

Compare: Product_{n>=1} (1-x^n)^3 = Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)/2).

Examples

			G.f.: A(x) = 1 + 6*x - 12*x^2 - 6*x^3 + 6*x^4 - 12*x^5 - 12*x^6 + 96*x^7 + 24*x^8 - 134*x^9 - 192*x^10 + 114*x^11 + 736*x^12 + 282*x^13 - 792*x^14 - 1532*x^15 - 270*x^16 + 1932*x^17 + 2004*x^18 + ...
RELATED SERIES.
The product P(x,y) = Product_{n>=1} (1 - (x^n + y^n))^3 begins
P(x,y) = 1 + (-3*x - 3*y) + (0*x^2 + 6*x*y + 0*y^2) + (5*x^3 + 6*x^2*y + 6*x*y^2 + 5*y^3) + (0*x^4 - 9*x^3*y - 12*x^2*y^2 - 9*x*y^3 + 0*y^4) + (0*x^5 - 9*x^4*y - 6*x^3*y^2 - 6*x^2*y^3 - 9*x*y^4 + 0*y^5) + (-7*x^6 - 9*x^5*y + 6*x^4*y^2 - 6*x^3*y^3 + 6*x^2*y^4 - 9*x*y^5 - 7*y^6) + (0*x^7 + 12*x^6*y + 12*x^5*y^2 + 27*x^4*y^3 + 27*x^3*y^4 + 12*x^2*y^5 + 12*x*y^6 + 0*y^7) + (0*x^8 + 12*x^7*y + 24*x^6*y^2 + 30*x^5*y^3 + 6*x^4*y^4 + 30*x^3*y^5 + 24*x^2*y^6 + 12*x*y^7 + 0*y^8) + (0*x^9 + 12*x^8*y - 12*x^7*y^2 - 23*x^6*y^3 - 24*x^5*y^4 - 24*x^4*y^5 - 23*x^3*y^6 - 12*x^2*y^7 + 12*x*y^8 + 0*y^9) + (9*x^10 + 12*x^9*y + 0*x^8*y^2 + 3*x^7*y^3 - 15*x^6*y^4 - 12*x^5*y^5 - 15*x^4*y^6 + 3*x^3*y^7 + 0*x^2*y^8 + 12*x*y^9 + 9*x^0*y^10) + (0*x^11 - 15*x^10*y - 36*x^9*y^2 - 54*x^8*y^3 - 60*x^7*y^4 - 60*x^6*y^5 - 60*x^5*y^6 - 60*x^4*y^7 - 54*x^3*y^8 - 36*x^2*y^9 - 15*x*y^10 + 0*y^11) + (0*x^12 - 15*x^11*y - 24*x^10*y^2 - 23*x^9*y^3 - 30*x^8*y^4 - 9*x^7*y^5 - 12*x^6*y^6 - 9*x^5*y^7 - 30*x^4*y^8 - 23*x^3*y^9 - 24*x^2*y^10 - 15*x*y^11 + 0*y^12) + (0*x^13 - 15*x^12*y - 6*x^11*y^2 - 12*x^10*y^3 + 51*x^9*y^4 + 57*x^8*y^5 + 54*x^7*y^6 + 54*x^6*y^7 + 57*x^5*y^8 + 51*x^4*y^9 - 12*x^3*y^10 - 6*x^2*y^11 - 15*x*y^12 + 0*y^13) + (0*x^14 - 15*x^13*y + 6*x^12*y^2 + 24*x^11*y^3 + 66*x^10*y^4 + 33*x^9*y^5 + 69*x^8*y^6 + 96*x^7*y^7 + 69*x^6*y^8 + 33*x^5*y^9 + 66*x^4*y^10 + 24*x^3*y^11 + 6*x^2*y^12 - 15*x*y^13 + 0*y^14) + (-11*x^15 - 15*x^14*y + 24*x^13*y^2 + 49*x^12*y^3 + 87*x^11*y^4 + 69*x^10*y^5 + 127*x^9*y^6 + 93*x^8*y^7 + 93*x^7*y^8 + 127*x^6*y^9 + 69*x^5*y^10 + 87*x^4*y^11 + 49*x^3*y^12 + 24*x^2*y^13 - 15*x*y^14 - 11*y^15) + ...
in which this sequence equals the coefficients of x^n*y^n for n >= 0.

Links

Crossrefs

Cf. A322210, A322211, A322213.

Programs

  • PARI
    {P = prod(n=1, 121, (1 - (x^n + y^n) +O(x^121) +O(y^121))^3 ); }
{a(n) = polcoeff( polcoeff( P, n, x), n, y)}
for(n=0, 120, print1( a(n), ", ") )

A322212 G.f.: P(x,y) = Product_{n>=1} (1 - (x^n + y^n)), where P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k, as a square table of coefficients T(n,k) read by antidiagonals.

Original entry on oeis.org

1, -1, -1, -1, 0, -1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 0, -1, -1, -2, -1, -1, -2, -1, -1, 0, 0, -1, 0, 0, 1, -2, 1, 0, 0, -1, 0, 0, -1, -1, 0, -1, 0, 0, -1, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, -1, 0, 0, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 0, 0, 0, 0, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 1, 0, 0, -1, 0, -1, 1, 0, -2, 0, 0, 0, 0, -2, 0, 1, -1, 0, -1, 0, 1, 1, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 1, 1, 2, 2, 2, -1, 1, 1, 4, 1, 1, -1, 2, 2, 2, 1, 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, -2, -2, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, -1, -2, -1, -2, -2, -2, -2, -2, -1, -2, -1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, -3, -1, 0, 0, 0, 0, -1, -3, 1, 1, 0, 0, 0, 1, 0
Offset: 0

Views

Author

Paul D. Hanna, Dec 04 2018

Keywords

Examples

			The product P(x,y) = Product_{n>=1} (1 - (x^n + y^n)) begins
P(x,y) = 1 +  (-1*x - 1*y) + (-1*x^2 + 0*x*y - 1*y^2) + (0*x^3 + 1*x^2*y + 1*x*y^2 + 0*y^3) + (0*x^4 + 1*x^3*y + 0*x^2*y^2 + 1*x*y^3 + 0*y^4) + (1*x^5 + 1*x^4*y + 1*x^3*y^2 + 1*x^2*y^3 + 1*x*y^4 + 1*y^5) + (0*x^6 + 0*x^5*y + 0*x^4*y^2 - 2*x^3*y^3 + 0*x^2*y^4 + 0*x*y^5 + 0*y^6) + (1*x^7 + 0*x^6*y + 0*x^5*y^2 + 0*x^4*y^3 + 0*x^3*y^4 + 0*x^2*y^5 + 0*x*y^6 + 1*y^7) + (0*x^8 - 1*x^7*y + 0*x^6*y^2 - 1*x^5*y^3 - 2*x^4*y^4 - 1*x^3*y^5 + 0*x^2*y^6 - 1*x*y^7 + 0*y^8) + (0*x^9 - 1*x^8*y - 1*x^7*y^2 - 2*x^6*y^3 - 1*x^5*y^4 - 1*x^4*y^5 - 2*x^3*y^6 - 1*x^2*y^7 - 1*x*y^8 + 0*y^9) + (0*x^10 - 1*x^9*y + 0*x^8*y^2 + 0*x^7*y^3 + 1*x^6*y^4 - 2*x^5*y^5 + 1*x^4*y^6 + 0*x^3*y^7 + 0*x^2*y^8 - 1*x*y^9 + 0*y^10) + (0*x^11 - 1*x^10*y - 1*x^9*y^2 + 0*x^8*y^3 - 1*x^7*y^4 + 0*x^6*y^5 + 0*x^5*y^6 - 1*x^4*y^7 + 0*x^3*y^8 - 1*x^2*y^9 - 1*x*y^10 + 0*y^11) + (-1*x^12 - 1*x^11*y + 0*x^10*y^2 + 0*x^9*y^3 - 1*x^8*y^4 + 0*x^7*y^5 + 0*x^6*y^6 + 0*x^5*y^7 - 1*x^4*y^8 + 0*x^3*y^9 + 0*x^2*y^10 - 1*x*y^11 - 1*y^12) + (0*x^13 + 0*x^12*y - 1*x^11*y^2 + 1*x^10*y^3 + 1*x^9*y^4 + 1*x^8*y^5 + 1*x^7*y^6 + 1*x^6*y^7 + 1*x^5*y^8 + 1*x^4*y^9 + 1*x^3*y^10 - 1*x^2*y^11 + 0*x*y^12 + 0*y^13) + (0*x^14 + 0*x^13*y + 1*x^12*y^2 + 2*x^11*y^3 + 2*x^10*y^4 + 2*x^9*y^5 + 3*x^8*y^6 + 2*x^7*y^7 + 3*x^6*y^8 + 2*x^5*y^9 + 2*x^4*y^10 + 2*x^3*y^11 + 1*x^2*y^12 + 0*x*y^13 + 0*y^14) + (-1*x^15 + 0*x^14*y - 1*x^13*y^2 + 1*x^12*y^3 + 0*x^11*y^4 - 2*x^10*y^5 + 0*x^9*y^6 + 0*x^8*y^7 + 0*x^7*y^8 + 0*x^6*y^9 - 2*x^5*y^10 + 0*x^4*y^11 + 1*x^3*y^12 - 1*x^2*y^13 + 0*x*y^14 - 1*y^15) + ...
This square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins
1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, ...;
-1, 0, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 1, ...;
-1, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, -1, 1, -1, 1, 0, ...;
0, 1, 1, -2, 0, -1, -2, 0, 0, 0, 1, 2, 1, 2, 1, 2, ...;
0, 1, 0, 0, -2, -1, 1, -1, -1, 1, 2, 0, 1, 1, 2, 0, ...;
1, 0, 0, -1, -1, -2, 0, 0, 1, 2, -2, 3, 2, 2, 0, -1, ...;
0, 0, 0, -2, 1, 0, 0, 1, 3, 0, 2, 0, -1, -1, -2, 1, ...;
1, -1, -1, 0, -1, 0, 1, 2, 0, 2, 1, 1, 0, -1, -3, 0, ...;
0, -1, 0, 0, -1, 1, 3, 0, 2, 1, 1, -1, -2, -1, -1, -3, ...;
0, -1, -1, 0, 1, 2, 0, 2, 1, 4, -2, -2, 0, -3, -3, -2, ...;
0, -1, 0, 1, 2, -2, 2, 1, 1, -2, -2, 0, -2, -2, -3, -4, ...;
0, -1, -1, 2, 0, 3, 0, 1, -1, -2, 0, 2, -5, -4, -2, -1, ...;
-1, 0, 1, 1, 1, 2, -1, 0, -2, 0, -2, -5, 0, -3, -2, 4, ...;
0, 0, -1, 2, 1, 2, -1, -1, -1, -3, -2, -4, -3, 4, 1, -5, ...;
0, 0, 1, 1, 2, 0, -2, -3, -1, -3, -3, -2, -2, 1, -2, 4, ...;
-1, 1, 0, 2, 0, -1, 1, 0, -3, -2, -4, -1, 4, -5, 4, 4, ...; ...
Alternatively, this sequence can be written as a triangle, starting as
1;
-1, -1;
-1, 0, -1;
0, 1, 1, 0;
0, 1, 0, 1, 0;
1, 1, 1, 1, 1, 1;
0, 0, 0, -2, 0, 0, 0;
1, 0, 0, 0, 0, 0, 0, 1;
0, -1, 0, -1, -2, -1, 0, -1, 0;
0, -1, -1, -2, -1, -1, -2, -1, -1, 0;
0, -1, 0, 0, 1, -2, 1, 0, 0, -1, 0;
0, -1, -1, 0, -1, 0, 0, -1, 0, -1, -1, 0;
-1, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, -1;
0, 0, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 0, 0;
0, 0, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 1, 0, 0;
-1, 0, -1, 1, 0, -2, 0, 0, 0, 0, -2, 0, 1, -1, 0, -1;
0, 1, 1, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 1, 1, 0;
0, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 0;
0, 1, 1, 2, 2, 2, -1, 1, 1, 4, 1, 1, -1, 2, 2, 2, 1, 1, 0;
0, 1, 0, 0, 0, 0, -1, 0, -1, -2, -2, -1, 0, -1, 0, 0, 0, 0, 1, 0;
0, 1, 1, 0, 0, -1, -2, -1, -2, -2, -2, -2, -2, -1, -2, -1, 0, 0, 1, 1, 0;
0, 1, 0, 0, 0, 1, 1, -3, -1, 0, 0, 0, 0, -1, -3, 1, 1, 0, 0, 0, 1, 0;
1, 1, 1, -1, 0, 1, -3, 0, -1, -3, -2, 2, -2, -3, -1, 0, -3, 1, 0, -1, 1, 1, 1;
0, 0, 0, -2, -1, -2, -3, -2, -3, -3, -2, -5, -5, -2, -3, -3, -2, -3, -2, -1, -2, 0, 0, 0;
0, 0, 0, -1, -1, -1, -4, -2, -5, -2, -3, -4, 0, -4, -3, -2, -5, -2, -4, -1, -1, -1, 0, 0, 0;
0, 0, 0, -3, -2, -3, -1, -2, -2, -3, -4, -2, -3, -3, -2, -4, -3, -2, -2, -1, -3, -2, -3, 0, 0, 0;
1, 0, 0, -2, -1, -1, -1, -1, -1, -3, 1, -1, -2, 4, -2, -1, 1, -3, -1, -1, -1, -1, -1, -2, 0, 0, 1; ...

Links

Crossrefs

Cf. A322213.

Programs

  • PARI
    {P = prod(n=1, 61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) ); }
{T(n, k) = polcoeff( polcoeff( P, n, x), k, y)}
for(n=0, 15, for(k=0, 15, print1( T(n, k), ", ") ); print(""))

A381011 a(n) = [(x*y)^n] Product_{k>=1} (1 - x^k - y^k)^k.

Original entry on oeis.org

1, 0, 2, -6, -14, -10, 32, 76, -80, -340, -200, 590, 2302, 1890, -3470, -11468, -16254, 5244, 57406, 109340, 81396, -158664, -550388, -829558, -359856, 1509570, 4333256, 6198660, 2628406, -10133230, -30439512, -46214582, -29696680, 45589368
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2025

Keywords

Crossrefs

Cf. A073592, A322213, A322214.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 - x^k - y^k)^k, {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 33}]

A381012 a(n) = [(x*y)^n] Product_{k>=1} (1 - x^k - y^k)^n.

Original entry on oeis.org

1, 0, -2, -6, -82, 530, -2420, 11718, -77458, 492834, -1022532, 3574714, -39670180, -172880396, 3186538080, -18558899356, 150869023214, -1286538054802, 6854805868780, -29675795883872, 168219184363308, -618102310289316, -1450440026397056, 26462673455854066
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2025

Keywords

Crossrefs

Cf. A008705, A322213, A322214.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 - x^k - y^k)^n, {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 23}]
Showing 1-5 of 5 results.

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