A384029 -id:A384029 - cp's OEIS frontend

A384031 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^4.

1, 4, 62, 1680, 65446, 3334800, 210218956, 15803243456, 1380404187558, 137419388080920, 15359405910256580, 1904647527097204032, 259511601503239509004, 38539384808775589973416, 6195988524478342471690200, 1072149116496356641327200000, 198683315255720972000976370950

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A129256, A384012, A384017.
Cf. A384060.
Cf. A382925, A384032, A384029, A351507.

Table[SeriesCoefficient[Product[(1+k*x)^4, {k, 1, n}], {x, 0, n}], {n, 0, 16}] (* Vaclav Kotesovec, May 18 2025 *)

a(n) = sum(i=0, n, sum(j=0, 3*n-i, sum(k=0, 3*n-i-j, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, k+1, 1)*stirling(n+1, 3*n-i-j-k+1, 1)))));

a(n) = Sum_{0<=i, j, k, l<=n and i+j+k+l=3*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1) * Stirling1(n+1,l+1)|.

a(n) ~ 2^(8*n + 7/2) * w^(4*n + 5/2) * n^(n - 1/2) / (sqrt(Pi*(w-1)) * 3^(3*n + 5/2) * exp(n) * (4*w-3)^n), where w = -LambertW(-1,-3*exp(-3/4)/4) = 1.300200741659068588153265179374583756429... - Vaclav Kotesovec, May 18 2025

A384018 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^3.

Original entry on oeis.org

1, 0, 3, 63, 1767, 63690, 2822740, 148810032, 9104502015, 634448680884, 49622704133175, 4305280182748875, 410376649359397380, 42633179822414174760, 4794685285831034253660, 580373328155358031572600, 75234419898396217903091151, 10398952352945773993329785448, 1526704288048697734221906020641

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A342111, A384029.

Cf. A384026.

a(n) = sum(i=0, n, sum(j=0, 2*n-i, abs(stirling(n, i, 1)*stirling(n, j, 1)*stirling(n, 2*n-i-j, 1))));

a(n) = Sum_{0<=i, j, k<=n and i+j+k=2*n} |Stirling1(n,i) * Stirling1(n,j) * Stirling1(n,k)|.

A384027 a(n) = [x^(3n)] Product_{k=0..n-1} (1 + kx)^4.

Original entry on oeis.org

1, 0, 0, 0, 1296, 2764800, 8041766400, 34726710251520, 219045033712578816, 1956771788423009992704, 24009126017002632247173120, 393692515265172002272138690560, 8424620140673205407840209386541056, 230472036551670538296109810120063451136, 7917891968134805796965854747528387122954240

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A342111, A384026.

Cf. A384029, A384030.

a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, n-i-j, abs(stirling(n, i, 1)*stirling(n, j, 1)*stirling(n, k, 1)*stirling(n, n-i-j-k, 1)))));

a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=n} |Stirling1(n,i) * Stirling1(n,j) * Stirling1(n,k) * Stirling1(n,l)|.

A384030 a(n) = [x^(2n)] Product_{k=0..n-1} (1 + kx)^4.

Original entry on oeis.org

1, 0, 1, 248, 79441, 38878520, 27741179521, 27412462941136, 35965398129639713, 60588665662486807184, 127588718827126433989569, 328596587850349392471155720, 1016488989627693108972046560497, 3720090951049096346043302894560648, 15901046580509525131539058273675597889

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A384027, A384029.

a(n) = sum(i=0, n, sum(j=0, 2*n-i, sum(k=0, 2*n-i-j, abs(stirling(n, i, 1)*stirling(n, j, 1)*stirling(n, k, 1)*stirling(n, 2*n-i-j-k, 1)))));

a(n) = Sum_{0<=i, j, k, l<=n and i+j+k+l=2*n} |Stirling1(n,i) * Stirling1(n,j) * Stirling1(n,k) * Stirling1(n,l)|.

A384089 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^n.

Original entry on oeis.org

1, 0, 1, 63, 7206, 1357300, 384271700, 153027592116, 81648987014364, 56259916067074896, 48646018448463951450, 51584263505394472459750, 65833976467770842558152992, 99553004175105699906002335098, 176031670802373999913671973955080, 359870756416991348769957239299854000

Offset: 0

Seiichi Manyama, May 19 2025

nonn

Cf. A342111, A384018, A384029.

Cf. A351507.

Table[SeriesCoefficient[Product[(1 + k*x)^n, {k, 0, n-1}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 19 2025 *)

a(n) = polcoef(prod(k=0, n-1, 1+k*x)^n, n);

a(n) = Sum_{0 <= x_1, x_2,..., x_n <= n and x_1 + x_2 + ... + x_n = (n-1)*n} Product_{k=1..n} |Stirling1(n,x_k)|.

a(n) ~ exp(n - 5/3) * n^(2*n+1) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). - Vaclav Kotesovec, May 19 2025

cp's OEIS Frontend

A384031 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^4.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A384018 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A384027 a(n) = [x^(3*n)] Product_{k=0..n-1} (1 + k*x)^4.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A384030 a(n) = [x^(2*n)] Product_{k=0..n-1} (1 + k*x)^4.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A384089 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A384027 a(n) = [x^(3n)] Product_{k=0..n-1} (1 + kx)^4.

A384030 a(n) = [x^(2n)] Product_{k=0..n-1} (1 + kx)^4.