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 1-4 of 4 results.

A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 7, 7, 1, 0, 1, 1, 15, 31, 19, 1, 0, 1, 1, 31, 115, 175, 51, 1, 0, 1, 1, 63, 391, 1255, 991, 141, 1, 0, 1, 1, 127, 1267, 8071, 13671, 5881, 393, 1, 0, 1, 1, 255, 3991, 49399, 161671, 160461, 35617, 1107, 1, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 27 2019

Keywords

Comments

T(n,k) is the constant term in the expansion of (-1 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.
For fixed k > 0, T(n,k) ~ (2^k - 1)^(n + (k-1)/2) / (2^((k-1)^2/2) * sqrt(k) * (Pi*n)^((k-1)/2)). - Vaclav Kotesovec, Oct 28 2019

Examples

			Square array begins:
   1, 1,  1,   1,     1,      1, ...
   1, 1,  1,   1,     1,      1, ...
   0, 1,  3,   7,    15,     31, ...
   0, 1,  7,  31,   115,    391, ...
   0, 1, 19, 175,  1255,   8071, ...
   0, 1, 51, 991, 13671, 161671, ...

Links

Crossrefs

Columns k=0..5 give A019590(n+1), A000012, A002426, A172634, A328725, A328750.
Main diagonal gives A328811.
T(n,n+1) gives A328813.
Cf. A309010, A328748, A328807.

Programs

  • Mathematica
    T[n_, k_] := Sum[(-1)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

A328725 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.

Original entry on oeis.org

1, 1, 15, 115, 1255, 13671, 160461, 1936425, 24071895, 305313415, 3939158905, 51521082405, 681635916325, 9105864515125, 122657982366375, 1664151758259915, 22720725637684215, 311933068664333175, 4303704125389134825, 59640225721889127525, 829774531966386480705
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2019

Keywords

Links

Crossrefs

Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^m: A002426 (m=2), A172634 (m=3), this sequence (m=4), A328750 (m=5).
Cf. A002899, A005260, A328713.

Programs

  • PARI
    {a(n) = polcoef(polcoef(polcoef((-1+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}
  • PARI
    {a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n,i)*sum(j=0, i, binomial(i, j)^4))}

Formula

a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^3*a(n) = (2*n - 1)^3*a(n-1) + (n-1)*(94*n^2 - 188*n + 93)*a(n-2) + 80*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 75*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 15^(n + 3/2) / (2^(11/2) * Pi^(3/2) * n^(3/2)). (End)

A328751 Constant term in the expansion of (-2 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

Original entry on oeis.org

1, 0, 30, 300, 6690, 124920, 2778600, 61790400, 1452751650, 34806097200, 855836532180, 21393889763400, 543342862524000, 13972938142363200, 363356617578926400, 9538720137580233600, 252510537115100657250, 6733792260826534332000, 180751978201192700659500
Offset: 0

Views

Author

Seiichi Manyama, Oct 27 2019

Keywords

Crossrefs

Column k=5 of A328748.
Cf. A005261, A328750.

Programs

  • Mathematica
    Table[Sum[(-2)^(n-i)*Binomial[n,i] * Sum[Binomial[i,j]^5, {j,0,i}], {i,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 20 2023 *)
  • PARI
    {a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}

Formula

a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^4*(22*n^2 - 198*n + 323)*a(n) = (n-1)*(198*n^5 - 1980*n^4 + 4535*n^3 - 2641*n^2 + 119*n + 210)*a(n-1) + (11066*n^6 - 143858*n^5 + 628715*n^4 - 1298438*n^3 + 1394723*n^2 - 756728*n + 165060)*a(n-2) + 4*(n-2)*(19096*n^5 - 248248*n^4 + 1086158*n^3 - 2156993*n^2 + 2004912*n - 708435)*a(n-3) + 40*(n-3)*(n-2)*(5346*n^4 - 64152*n^3 + 242653*n^2 - 363566*n + 182959)*a(n-4) + 400*(n-4)*(n-3)*(n-2)*(682*n^3 - 6820*n^2 + 17955*n - 12432)*a(n-5) + 6000*(n-5)*(n-4)*(n-3)*(n-2)*(22*n^2 - 154*n + 147)*a(n-6).
a(n) ~ 2^(n-6) * 3^(n+2) * 5^(n + 3/2) / (Pi^2 * n^2). (End)

A328809 Constant term in the expansion of (1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

Original entry on oeis.org

1, 3, 39, 597, 11991, 260613, 6129489, 151078707, 3867441111, 101852866533, 2744610170049, 75348380209347, 2100889194001761, 59349600029522403, 1695505948476461559, 48909452234258070117, 1422877722974198091351, 41704912707174877940613
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2019

Keywords

Crossrefs

Column k=5 of A328807.
Cf. A005261, A328750, A328751.

Programs

  • Mathematica
    Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^5, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)
  • PARI
    {a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}

Formula

a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^4*(40*n^2 - 24*n - 79)*a(n) = (1080*n^6 - 2808*n^5 + 875*n^4 + 2928*n^3 - 3762*n^2 + 1834*n - 336)*a(n-1) + (9320*n^6 - 42872*n^5 + 61193*n^4 - 12152*n^3 - 35518*n^2 + 21658*n - 2016)*a(n-2) - (n-2)*(48560*n^5 - 223376*n^4 + 216118*n^3 + 381866*n^2 - 791133*n + 355194)*a(n-3) + (n-3)*(n-2)*(79560*n^4 - 286416*n^3 - 56675*n^2 + 976675*n - 616322)*a(n-4) - 11*(n-4)*(n-3)*(n-2)*(5080*n^3 - 8128*n^2 - 25641*n + 21693)*a(n-5) + 363*(n-5)*(n-4)*(n-3)*(n-2)*(40*n^2 + 56*n - 63)*a(n-6).
a(n) ~ 33^(n+2) / (256 * sqrt(5) * Pi^2 * n^2). (End)
Showing 1-4 of 4 results.

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