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.

A328748 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 1, 0, 0, 2, 1, 0, 2, 0, -3, 1, 0, 6, 0, 0, 4, 1, 0, 14, 12, 6, 0, -5, 1, 0, 30, 72, 90, 0, 0, 6, 1, 0, 62, 300, 882, 360, 20, 0, -7, 1, 0, 126, 1080, 6690, 8400, 2040, 0, 0, 8, 1, 0, 254, 3612, 44706, 124920, 95180, 10080, 70, 0, -9
Offset: 0

Views

Author

Seiichi Manyama, Oct 27 2019

Keywords

Comments

T(n,k) is the constant term in the expansion of (-2 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.

Examples

			Square array begins:
    1, 1, 1,   1,    1,      1, ...
    0, 0, 0,   0,    0,      0, ...
   -1, 0, 2,   6,   14,     30, ...
    2, 0, 0,  12,   72,    300, ...
   -3, 0, 6,  90,  882,   6690, ...
    4, 0, 0, 360, 8400, 124920, ...

Links

Crossrefs

Columns k=0..5 give A097141(n+1), A000007, A126869, A002898, A328735, A328751.
T(n,n+1) gives A328814.
Cf. A309010, A328747, A328807.

Programs

  • Mathematica
    T[n_, k_] := Sum[(-2)^(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 *)

A328750 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, 1, 31, 391, 8071, 161671, 3634921, 84109201, 2032357111, 50355327991, 1277302604521, 32983865502721, 864982811998801, 22976755021842961, 617140285389771391, 16735405610179740151, 457647302453165769751, 12607719926638032161431, 349620344754345216824041
Offset: 0

Views

Author

Seiichi Manyama, Oct 27 2019

Keywords

Crossrefs

Column k=5 of A328747.
Cf. A005261, A328751.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - i)*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, (-1)^(n-i)*binomial(n,i)*sum(j=0, i, binomial(i, j)^5))}

Formula

a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^4*(440*n^2 - 2728*n + 3723)*a(n) = (6600*n^6 - 54120*n^5 + 147013*n^4 - 174348*n^3 + 102442*n^2 - 29260*n + 3108)*a(n-1) + (194920*n^6 - 1988184*n^5 + 7650713*n^4 - 14588908*n^3 + 14793198*n^2 - 7658420*n + 1601964)*a(n-2) + (n-2)*(690800*n^5 - 7046160*n^4 + 26712814*n^3 - 47822370*n^2 + 40779795*n - 13361628)*a(n-3) + (n-3)*(n-2)*(975480*n^4 - 8974416*n^3 + 28602923*n^2 - 37477643*n + 16905924)*a(n-4) + (n-4)*(n-3)*(n-2)*(622600*n^3 - 4482720*n^2 + 9455173*n - 5628497)*a(n-5) + 341*(n-5)*(n-4)*(n-3)*(n-2)*(440*n^2 - 1848*n + 1435)*a(n-6).
a(n) ~ 31^(n+2) / (256 * sqrt(5) * Pi^2 * n^2). (End)

A328735 Constant term in the expansion of (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, 0, 14, 72, 882, 8400, 95180, 1060080, 12389650, 146472480, 1767391164, 21581516880, 266718438756, 3327025429728, 41849031952728, 530135326392672, 6757845419895570, 86619827323917888, 1115719258312182524, 14434274832755201424, 187477238295444829732
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2019

Keywords

Crossrefs

Column k=4 of A328748.
Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^m: A126869 (m=2), A002898 (m=3), this sequence (m=4), A328751 (m=5).
Cf. A002899, A005260, A328725.

Programs

  • Mathematica
    Table[Sum[(-2)^(n-i)*Binomial[n,i] * Sum[Binomial[i,j]^4, {j,0,i}], {i,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 20 2023 *)
  • PARI
    {a(n) = polcoef(polcoef(polcoef((-2+(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, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

Formula

a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^3*a(n) = 2*(n-1)*n*(2*n - 1)*a(n-1) + 112*(n-1)^3*a(n-2) + 184*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 336*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(n-4) * 7^(n + 3/2) / (Pi^(3/2) * n^(3/2)). (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.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.