A322211 -id:A322211 - cp's OEIS frontend

A322210 G.f.: P(x,y) = Product_{n>=1} 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.

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 7, 10, 7, 5, 7, 12, 18, 18, 12, 7, 11, 19, 34, 38, 34, 19, 11, 15, 30, 56, 74, 74, 56, 30, 15, 22, 45, 94, 133, 158, 133, 94, 45, 22, 30, 67, 146, 233, 297, 297, 233, 146, 67, 30, 42, 97, 228, 385, 550, 602, 550, 385, 228, 97, 42, 56, 139, 340, 623, 951, 1166, 1166, 951, 623, 340, 139, 56

Offset: 0

Paul D. Hanna, Nov 30 2018

Conjecture 1: the triangular table T(n,k) is the number of ways to form the subsum k from the partitions of n, where n and k are integers such that 0 <= k <= n. For example, t(4,2)=10; the five partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1) with subsum 2 occurring {0,0,2,2,6) times for a total of 10. - George Beck, Jan 03 2020
From Wouter Meeussen, Mar 09 2023: (Start)
Conjecture 2: the square table T(n,k) is the coefficient of s_lambda in the sum over all partitions lambda |-n and nu |-k of (s_rho/mu) where s_lambda*s_mu = Sum(rho|-n+k; C(rho, lambda, mu) s_rho). Simply stated as: multiply lambda with mu, and, for each term in the result, take the skew Schur function with mu and count how often you get the original lambda back. Sum up over all lambda and mu of the size n and k.
Conjecture 3: the triangular table T(n,k) is analogous to conjecture 2, but counting s_lambda in s_(lambda/mu) * s_mu with lambda |- n and mu |- k and 0<=k<=n. (End)

			G.f.: 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) + ...
such that
P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)),
where
P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k.
SQUARE TABLE.
The square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins
   1,   1,   2,    3,    5,     7,    11,     15,     22,     30, ...
   1,   2,   4,    7,   12,    19,    30,     45,     67,     97, ...
   2,   4,  10,   18,   34,    56,    94,    146,    228,    340, ...
   3,   7,  18,   38,   74,   133,   233,    385,    623,    977, ...
   5,  12,  34,   74,  158,   297,   550,    951,   1614,   2627, ...
   7,  19,  56,  133,  297,   602,  1166,   2133,   3775,   6437, ...
  11,  30,  94,  233,  550,  1166,  2382,   4551,   8424,  14953, ...
  15,  45, 146,  385,  951,  2133,  4551,   9142,  17639,  32680, ...
  22,  67, 228,  623, 1614,  3775,  8424,  17639,  35492,  68356, ...
  30,  97, 340,  977, 2627,  6437, 14953,  32680,  68356, 136936, ...
  42, 139, 506, 1501, 4202, 10692, 25835,  58659, 127443, 264747, ...
  56, 195, 730, 2255, 6531, 17290, 43313, 102149, 229998, 495195, ...
  ...
TRIANGLE.
Alternatively, this sequence may be written as a triangle, starting as
   1;
   1,   1;
   2,   2,   2;
   3,   4,   4,   3;
   5,   7,  10,   7,    5;
   7,  12,  18,  18,   12,    7;
  11,  19,  34,  38,   34,   19,   11;
  15,  30,  56,  74,   74,   56,   30,   15;
  22,  45,  94, 133,  158,  133,   94,   45,   22;
  30,  67, 146, 233,  297,  297,  233,  146,   67,  30;
  42,  97, 228, 385,  550,  602,  550,  385,  228,  97,  42;
  56, 139, 340, 623,  951, 1166, 1166,  951,  623, 340, 139,  56;
  77, 195, 506, 977, 1614, 2133, 2382, 2133, 1614, 977, 506, 195, 77;
  ...

Alois P. Heinz, Antidiagonals n = 0..200 (first 61 antidiagonals from Paul D. Hanna)

Cf. A322200 (log).
Cf. A000041 (row 0 = partitions), A000070 (row 1), A093695(k+2) (row 2).
Main diagonal gives A322211.
Antidiagonal sums give A070933.
Cf. A284593.
Cf. A361286.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1,
      (x+1)^n, b(n, i-1) +(x^i+1)*b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 23 2019

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + (x^i + 1) b[n - i, Min[n - i, i]]]];
T[n_, k_] := Coefficient[b[n + k, n + k], x, k];
Table[Table[T[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

{P = 1/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=0,16, for(k=0,16, print1( T(n,k),", ") );print(""))

FORMULAS FOR TERMS.
T(n,k) = T(k,n) for n >= 0, k >= 0.
T(n,0) = A000041(n) for n >= 0, where A000041 is the partition numbers.
T(n,1) = A000070(n) for n >= 0, where A000070 is the sum of partitions.
ROW GENERATING FUNCTIONS.
Row 0: 1/( Product_{n>=1} (1 - x^n) ).
Row 1: 1/( (1-x) * Product_{n>=1} (1 - x^n) ).
Row 2: 2/( (1-x) * (1-x^2) * Product_{n>=1} (1 - x^n) ).

A322213 a(n) = coefficient of x^n*y^n in Product_{n>=1} (1 - (x^n + y^n)).

Original entry on oeis.org

1, 0, 0, -2, -2, -2, 0, 2, 2, 4, -2, 2, 0, 4, -2, 4, 6, 4, 4, -2, -10, -12, -12, -16, -14, 6, 6, -4, 16, 22, 30, 12, 18, -60, -18, -34, -64, -56, -36, -46, 16, 46, 64, 70, 110, 192, 152, 124, 58, 78, -26, -54, -366, -278, -182, -282, -190, 40, -112, 234, 300, 476, 488, 906, 732, 616, 706, 154, 228, -180, -864, -1112, -1744, -2294, -2824, -3154, -2170, -2146, -2524, -1102, -476, -126, 1986, 4338, 3344, 3608, 6316, 5136, 6638, 6726, 5254, 3982, 2916, -1466, -86, -6710, -6502, -9900, -9128, -14170, -12232, -13940, -9192, -6892, -6270, 3762, 7058, 9468, 23860, 22556, 29812, 40150, 34952, 30350

Offset: 0

Paul D. Hanna, Dec 03 2018

sign

			G.f.: A(x) = 1 - 2*x^3 - 2*x^4 - 2*x^5 + 2*x^7 + 2*x^8 + 4*x^9 - 2*x^10 + 2*x^11 + 4*x^13 - 2*x^14 + 4*x^15 + 6*x^16 + 4*x^17 + 4*x^18 - 2*x^19 - 10*x^20 + ...
RELATED SERIES.
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) + ...
in which this sequence equals the coefficients of x^n*y^n for n >= 0.

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A322211.

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

A292613 a(n) = [x^n] 1/(1-x)^n * Product_{k=1..n} 1/(1-x^k).

Original entry on oeis.org

1, 2, 7, 25, 92, 343, 1292, 4902, 18703, 71677, 275694, 1063636, 4114131, 15948762, 61946290, 241013869, 939125870, 3664299332, 14314777054, 55982787136, 219158088711, 858728875776, 3367576480747, 13216392846128, 51905939548950, 203989227456894, 802164259099114

Offset: 0

Vaclav Kotesovec, Sep 20 2017

nonn

Number of ways to pick n units in all partitions of 2n - Olivier Gérard, May 07 2020

			Illustration of comment for n=3, a(3)=25 :
Among the 11 integer partitions of 6, 3 have at least 3 ones.
3,1,1,1  ;  2,1,1,1,1;  1,1,1,1,1,1;
There are respectively 1, 4 and 20 ways to pick 3 of these.

Alois P. Heinz, Table of n, a(n) for n = 0..1663 (first 301 terms from Vaclav Kotesovec)

Cf. A000041, A001700, A014329, A292463.

Cf. A292508, A322211.

Table[SeriesCoefficient[1/(1-x)^n*Product[1/(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]

a(n) ~ c * 4^n / sqrt(Pi*n), where c = 1/(2*QPochhammer[1/2, 1/2]) = 1.7313733097275318057689... - Vaclav Kotesovec, Sep 20 2017

a(n) = A292508(n,n+1). - Alois P. Heinz, Jul 16 2021

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

Paul D. Hanna, Dec 04 2018

sign

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

			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.

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A322210, A322211, A322213.

{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), ", ") )

A322198 a(n) is the coefficient of x^ny^n in Product_{n>=1} 1/(1 - x^(2n-1) - y^(2*n-1)).

Original entry on oeis.org

1, 2, 6, 24, 84, 312, 1174, 4420, 16772, 64014, 245212, 942668, 3634914, 14051530, 54440336, 211331906, 821779372, 3200447054, 12481364146, 48736064248, 190513382908, 745492958862, 2919891150694, 11446207136530, 44905452622268, 176300343498632, 692629144937724, 2722834581642342, 10710164125130394, 42151077430686344, 165975440541202824, 653864689092828458

Offset: 0

Paul D. Hanna, Dec 07 2018

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 84*x^4 + 312*x^5 + 1174*x^6 + 4420*x^7 + 16772*x^8 + 64014*x^9 + 245212*x^10 + 942668*x^11 + 3634914*x^12 + ...
RELATED SERIES.
The product P(x,y) = 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) * ...)
may be expressed as the series expansion
P(x,y) = 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) + ...
in which this sequence equals the coefficients of x^n*y^n for n >= 0.

Paul D. Hanna, Table of n, a(n) for n = 0..512

Cf. A322187, A322211.

N=35;
{P = 1/prod(n=1, N+1, (1 - x^(2*n-1) - y^(2*n-1) +x^2*O(x^N) +y^2*O(y^N)) ); }
{a(n) = polcoeff( polcoeff( P, n, x), n, y)}
for(n=0, N, print1( a(n), ", ") )

a(n) ~ c * 4^n / sqrt(n), where c = 1/(sqrt(Pi) * QPochhammer(1/4)) = 0.819402796697705077405540985476846791094716961849197... - Vaclav Kotesovec, Jun 18 2019, updated Mar 17 2024

A382979 a(n) = [(x*y)^n] Product_{k>=1} 1/(1 - x^k + y^k).

Original entry on oeis.org

1, -2, 4, -20, 78, -282, 1048, -4014, 15456, -59224, 227646, -879694, 3407730, -13219372, 51375286, -200021556, 779870542, -3044448644, 11898709560, -46553635346, 182315752476, -714619687038, 2803342734160, -11005274516610, 43233909672938, -169951684067602, 668474115081988

Offset: 0

Seiichi Manyama, Apr 11 2025

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Main diagonal of A382974.

Cf. A322211, A100221.

nmax := 26; prec := 2*nmax + 10; Rx := PowerSeriesRing(Rationals(), prec); Rxy := PowerSeriesRing(Rx, prec); P := Rxy!1; for k in [1..prec] do P *:= 1/(1 - x^k + y^k); end for; seq := [Coefficient(Coefficient(P, n), n) : n in [0..nmax]]; print seq; // Vincenzo Librandi, Apr 12 2025

a[n_]:=SeriesCoefficient[Product[1/(1-x^k+y^k),{k,1,n+5}],{x,0,n},{y,0,n}]; Table[a[n],{n,0,26}] (* Vincenzo Librandi, Apr 12 2025 *)

a(n) ~ (-1)^n * 4^n / (A100221 * sqrt(Pi*n)). - Vaclav Kotesovec, Apr 13 2025

A382947 a(n) = [(x*y)^n] Product_{k>=1} 1 / (1 - x^k - y^k)^k.

Original entry on oeis.org

1, 2, 16, 78, 426, 1940, 9300, 40530, 177940, 749788, 3137352, 12865488, 52425432, 211336062, 848099898, 3385259588, 13475690578, 53504526568, 212146065506, 840218845230, 3325872415258, 13159945010474, 52064974607244, 205979887425498, 814961759722486

Offset: 0

Ilya Gutkovskiy, Apr 09 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000219, A322211.

Table[SeriesCoefficient[Product[1/(1 - x^k - y^k)^k, {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 24}]

a(n) ~ c * 4^n / sqrt(n), where c = 19.59922592... - Vaclav Kotesovec, Apr 11 2025

A382949 a(n) = [(x*y)^n] Product_{k>=1} 1 / (1 - x^k - y^k)^n.

Original entry on oeis.org

1, 2, 48, 1190, 33648, 996292, 30626316, 965163166, 30995087312, 1009925740946, 33289934968618, 1107728567917028, 37149902553751260, 1254165186821008126, 42580296599191705276, 1452739684287637542640, 49776378699192072523920, 1711962807156690517057454

Offset: 0

Ilya Gutkovskiy, Apr 09 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A008485, A322211.

Table[SeriesCoefficient[Product[1/(1 - x^k - y^k)^n, {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 17}]

a(n) ~ c * d^n / n, where d = 36.5023860624117446261380818... and c = 0.08167564464819257818345... - Vaclav Kotesovec, Apr 10 2025

A382958 a(n) = (n!)^2 * [(x*y)^n] Product_{k>=1} 1 / (1 - (x^k + y^k)/k!).

Original entry on oeis.org

1, 2, 30, 920, 53078, 4828892, 643086588, 117718532696, 28378716172822, 8713799596723484, 3320414836230009080, 1537509304647364575716, 850310874146059999520372, 553587598414859641796343780, 419087377790397643526857611312, 365040505934072220586791778761920

Offset: 0

Ilya Gutkovskiy, Apr 10 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A005651, A322211.

Table[(n!)^2 SeriesCoefficient[Product[1/(1 - (x^k + y^k)/k!), {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 15}]

a(n) ~ c * sqrt(Pi) * 2^(2*n + 1) * n^(2*n + 1/2) / exp(2*n), where c = Product_{k>=2} (1 + 1/(2^(k-1)*k! - 1)) = 1.399382837233736726730568376611759424994992988... - Vaclav Kotesovec, Apr 24 2025

cp's OEIS Frontend

A322210 G.f.: P(x,y) = Product_{n>=1} 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A322213 a(n) = coefficient of x^n*y^n in Product_{n>=1} (1 - (x^n + y^n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A292613 a(n) = [x^n] 1/(1-x)^n * Product_{k=1..n} 1/(1-x^k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A322198 a(n) is the coefficient of x^n*y^n in Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A382979 a(n) = [(x*y)^n] Product_{k>=1} 1/(1 - x^k + y^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A382947 a(n) = [(x*y)^n] Product_{k>=1} 1 / (1 - x^k - y^k)^k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A382949 a(n) = [(x*y)^n] Product_{k>=1} 1 / (1 - x^k - y^k)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

A322198 a(n) is the coefficient of x^ny^n in Product_{n>=1} 1/(1 - x^(2n-1) - y^(2*n-1)).