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.

A346764 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(3*k,k) / (2*k + 1).

Original entry on oeis.org

1, 1, 4, 22, 149, 1169, 10272, 99012, 1032346, 11526094, 136755650, 1714031312, 22584475206, 311597054110, 4486616619986, 67227958200996, 1045724188868353, 16849477086762701, 280694278424099214, 4826423610068933738, 85527389275821664161, 1559842051063534891301
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Comments

Stirling transform of A001764.

Links

Crossrefs

Cf. A001764, A064856, A346765, A346766, A346767, A346768, A346769.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n, k] Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; CoefficientList[Series[Sum[(Binomial[3 k, k]/(2 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[HypergeometricPFQ[{1/3, 2/3}, {1, 3/2}, 27 (Exp[x] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(3*k, k)/(2*k + 1)); \\ Michel Marcus, Aug 02 2021

Formula

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).

A346765 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(4*k,k) / (3*k + 1).

Original entry on oeis.org

1, 1, 5, 35, 301, 2980, 32824, 394119, 5089387, 70008606, 1018551386, 15586572831, 249761256325, 4175639393112, 72613795311014, 1310044170067051, 24465311302401475, 472024733580022982, 9392470260695334398, 192455730876780393835, 4055291189439769281557
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Comments

Stirling transform of A002293.

Links

Crossrefs

Cf. A002293, A064856, A346764, A346766, A346767, A346768, A346769.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n, k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Sum[(Binomial[4 k, k]/(3 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[HypergeometricPFQ[{1/4, 1/2, 3/4}, {2/3, 1, 4/3}, 256 (Exp[x] - 1)/27], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(4*k, k)/(3*k + 1)); \\ Michel Marcus, Aug 02 2021

Formula

G.f.: Sum_{k>=0} ( binomial(4*k,k) / (3*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).

A346767 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(6*k,k) / (5*k + 1).

Original entry on oeis.org

1, 1, 7, 70, 855, 11907, 182714, 3029040, 53565875, 1001599339, 19674910572, 404009742858, 8638256718929, 191702754433132, 4403979321915615, 104496256532120370, 2555972287817569101, 64340126437548435175, 1664318438781195696512, 44182488823505663971205
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Comments

Stirling transform of A002295.

Links

Crossrefs

Cf. A002295, A064856, A346764, A346765, A346766, A346768, A346769.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n, k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 19; CoefficientList[Series[HypergeometricPFQ[{1/6, 1/3, 1/2, 2/3, 5/6}, {2/5, 3/5, 4/5, 1, 6/5}, 46656 (Exp[x] - 1)/3125], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(6*k, k)/(5*k + 1)); \\ Michel Marcus, Aug 02 2021

Formula

G.f.: Sum_{k>=0} ( binomial(6*k,k) / (5*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).

A346768 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(7*k,k) / (6*k + 1).

Original entry on oeis.org

1, 1, 8, 92, 1289, 20518, 358611, 6749268, 135095116, 2851394415, 63066764910, 1454808403309, 34869538474423, 865771965143262, 22211885496614803, 587583912259110350, 15998031596388750905, 447598845624472993496, 12850922242548662924046, 378153449033278630907275
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Comments

Stirling transform of A002296.

Links

Crossrefs

Cf. A002296, A064856, A346764, A346765, A346766, A346767, A346769.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n, k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Sum[(Binomial[7 k, k]/(6 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 19; CoefficientList[Series[HypergeometricPFQ[{1/7, 2/7, 3/7, 4/7, 5/7, 6/7}, {1/3, 1/2, 2/3, 5/6,1, 7/6}, 823543 (Exp[x] - 1)/46656], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(7*k, k)/(6*k + 1)); \\ Michel Marcus, Aug 02 2021

Formula

G.f.: Sum_{k>=0} ( binomial(7*k,k) / (6*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).

A346766 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(5*k,k) / (4*k + 1).

Original entry on oeis.org

1, 1, 6, 51, 531, 6331, 83532, 1195452, 18316582, 297727712, 5099398853, 91554269703, 1715910362408, 33457504204403, 676778172939139, 14168046060375184, 306327815585165519, 6827996259530724139, 156654003923243040925, 3694188118839057258940, 89428870506038692255920
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Comments

Stirling transform of A002294.

Crossrefs

Cf. A002294, A064856, A346764, A346765, A346767, A346768, A346769.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n, k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Sum[(Binomial[5 k, k]/(4 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5}, {1/2, 3/4, 1, 5/4}, 3125 (Exp[x] - 1)/256], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(5*k, k)/(4*k + 1)); \\ Michel Marcus, Aug 03 2021

Formula

G.f.: Sum_{k>=0} ( binomial(5*k,k) / (4*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).
Showing 1-5 of 5 results.

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