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 1-10 of 13 results. Next

A086672 Stirling1 transform of Catalan numbers: Sum_{k=0..n} Stirling1(n,k)*binomial(2*k,k)/(k+1).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, -5, 29, -196, 1518, -13266, 129163, -1386572, 16270671, -207195495, 2845705719, -41930575740, 659781404944, -11041824881696, 195839234324062, -3669384701403344, 72423881548363354, -1501924519315744146, 32649768696532126439, -742432111781693213350
Offset: 0

Views

Author

Vladeta Jovovic, Sep 12 2003

Keywords

Comments

1, 1, 1, 0, 1, -5, 29, -196, ... is the Stirling1 transform of the Motzkin numbers A001006. - Philippe Deléham, May 27 2015

Links

Crossrefs

Cf. A000108, A001006, A008275, A048994, A064856.
Cf. A201950, A201952, A306335.

Programs

  • Mathematica
    Table[Sum[StirlingS1[n, k] * CatalanNumber[k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Aug 04 2021 *)
  • PARI
    a(n)={sum(k=0, n, stirling(n,k,1) * binomial(2*k, k) / (k+1))} \\ Andrew Howroyd, Jan 27 2020

Formula

E.g.f.: hypergeom([1/2], [2], 4*log(1+x)) = (1+x)^2*(BesselI(0, 2*log(1+x))-BesselI(1, 2*log(1+x))).
Let C(m) be the m-th Catalan number, A000108(m). Let S(m, n) = an unsigned Stirling number of the first kind. Then a(m) = sum{k=0 to m} S(m, k) C(k) (-1)^(k+m). - Leroy Quet, Jan 23 2004
E.g.f. f(x) satisfies f(x) = 1 + integral{0 to x} f(y) f((x-y)/(1+y))/(1+y) dy. - Leroy Quet, Jan 25 2004
a(n) = Sum_{k = 0..n} A048994(n, k) * A000108(k). - Philippe Deléham, May 27 2015
a(n+1) = Sum_{k = 0..n} A048994(n,k) * A001006(k). - Philippe Deléham, May 27 2015
For n > 1, a(n) = (A201950(n+1) - (3*n-2)*A201950(n) + n*(3*n-7)*A201950(n-1) - (n-4)*(n-1)*n*A201950(n-2)) * (-1)^n/2. - Vaclav Kotesovec, May 04 2024

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 27 2020

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).

A346769 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(8*k,k) / (7*k + 1).

Original entry on oeis.org

1, 1, 9, 117, 1849, 33099, 648683, 13652529, 304828941, 7160371928, 175882500852, 4497024667232, 119255943612372, 3270580645588057, 92537409967439493, 2695752129992788115, 80716475549045336327, 2480352681613911495046, 78120174740199126232258
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Comments

Stirling transform of A007556.

Links

Crossrefs

Cf. A007556, A064856, A346764, A346765, A346766, A346767, A346768.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n, k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Sum[(Binomial[8 k, k]/(7 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 18; CoefficientList[Series[HypergeometricPFQ[{1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8}, {2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7}, 16777216 (Exp[x] - 1)/823543], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(8*k, k)/(7*k + 1)); \\ Michel Marcus, Aug 02 2021

Formula

G.f.: Sum_{k>=0} ( binomial(8*k,k) / (7*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).

A086662 Stirling transform of Catalan numbers: Sum_{k=0..n} |Stirling1(n,k)|*C(2*k,k)/(k+1).

Original entry on oeis.org

1, 1, 3, 13, 72, 481, 3745, 33209, 329868, 3624270, 43608474, 570008803, 8039735704, 121673027607, 1966231022067, 33786076421499, 615043147866660, 11822938288619344, 239298079351004608, 5086498410027323134, 113278368771499790136, 2637549737582063583274, 64082443707327038140602, 1621782672366231029685407
Offset: 0

Views

Author

Vladeta Jovovic, Sep 12 2003

Keywords

Links

Crossrefs

Cf. A000108, A008275, A064856.

Programs

  • Mathematica
    CoefficientList[Series[(BesselI[0, 2*Log[1-x]] + BesselI[1, 2*Log[1-x]]) / (1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Mar 02 2014 *)
Table[Sum[Abs[StirlingS1[n,k]]*Binomial[2*k,k]/(k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Mar 02 2014 *)
  • PARI
    a(n)=sum(k=0,n, abs(stirling(n,k,1)) * binomial(2*k,k)/(k+1) ); \\ Joerg Arndt, Mar 02 2014

Formula

E.g.f.: hypergeom([1/2], [2], -4*log(1-x)) = 1/(1-x)^2*(BesselI(0, 2*log(1-x))+BesselI(1, 2*log(1-x))).
a(n)=(1/(2*pi))*int(product(x+k,k,0,n-1)*sqrt((4-x)/x),x,0,4) (moment representation). [Paul Barry, Jul 26 2010]

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).

A305406 Expansion of Sum_{k>=0} binomial(2*k,k)*x^k/Product_{j=1..k} (1 - j*x).

Original entry on oeis.org

1, 2, 8, 40, 234, 1544, 11242, 89016, 758504, 6900012, 66590782, 678322704, 7262393832, 81431657220, 953339019606, 11622207372104, 147199295291518, 1932876310310488, 26265519359529974, 368752956750812256, 5340795881536757632, 79691179458925839676, 1223524383429928039306
Offset: 0

Views

Author

Ilya Gutkovskiy, May 31 2018

Keywords

Comments

Stirling transform of A000984.

Links

Crossrefs

Cf. A000984, A064856.

Programs

  • Maple
    b:= proc(n, m) option remember;
     `if`(n=0, binomial(2*m, m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 04 2021
  • Mathematica
    nmax = 22; CoefficientList[Series[Sum[Binomial[2 k, k] x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Exp[2 (Exp[x] - 1)] BesselI[0, 2 (Exp[x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] Binomial[2 k, k], {k, 0, n}], {n, 0, 22}]

Formula

E.g.f.: exp(2*(exp(x) - 1))*BesselI(0,2*(exp(x) - 1)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(2*k,k).

A066053 Stirling transform of A002457.

Original entry on oeis.org

1, 6, 36, 236, 1686, 13028, 108078, 956348, 8976708, 88962160, 927129786, 10125636716, 115543526476, 1373933166848, 16985192456410, 217851008508220, 2893517713599370, 39732264695056772, 563187218351672330, 8229159647194683140, 123795221970087313340
Offset: 0

Views

Author

Karol A. Penson, Nov 30 2001

Keywords

Links

Crossrefs

Cf. A002457, A064856.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
      (2*m+1)!/m!^2, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 23 2023
  • Mathematica
    Table[Sum[StirlingS2[n, k]*(2*k+2)!/(2*k!*(k+1)!), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 01 2018 *)

Formula

a(n) = sum(stirling2(n, k)*(2*k + 2)!/(2*k!*(k + 1)!), k = 0..n), n = 0, 1, ...;
E.g.f.: exp(2*exp(x) - 2)*(BesselI(0, 2*exp(x) - 2) + 4*BesselI(0, 2*exp(x) - 2)*(exp(x) - 1) + 4*(exp(x) - 1)*BesselI(1, 2*exp(x) - 2)).
Showing 1-10 of 13 results. Next

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