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.

A037972 a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2.

Original entry on oeis.org

0, 1, 12, 108, 800, 5250, 31752, 181104, 988416, 5212350, 26741000, 134132856, 660284352, 3199016548, 15288882000, 72209880000, 337535723520, 1563410094390, 7182839945160, 32761238433000, 148450107960000, 668693511305820, 2995943329133040, 13356820221694560
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • Identity (3.78), S_{3}, in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 31.

Links

Crossrefs

Cf. A000108, A000217, A037965, A037966, A074334, A329913, A329444.

Programs

  • Magma
    [n^2*(n+1)*Binomial(2*n-2, n-1)/2: n in [0..30]]; // G. C. Greubel, Jun 22 2022
  • Mathematica
    Table[n^2*Binomial[n+1,2]*CatalanNumber[n-1], {n,0,30}] (* G. C. Greubel, Jun 22 2022 *)
  • PARI
    {a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2} \\ Seiichi Manyama, Aug 09 2020
  • SageMath
    [n^2*binomial(n+1,2)*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 22 2022

Formula

a(n) = Sum_{k=0..n} k^3*(C(n,k))^2. [heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010]
a(n) = A000217(n)*A037965(n). - R. J. Mathar, Jul 26 2015
(n-1)^2*a(n) = 2*(11*n-16)*a(n-1) + 8*n*(2*n-5)*a(n-2). - R. J. Mathar, Oct 20 2015
(n-1)^3*a(n) = 2*n*(n+1)*(2*n-3)*a(n-1). - R. J. Mathar, Oct 20 2015
G.f.: x * (1 - 2*x + 10*x^2 - 12*x^3) / (1 - 4*x)^(7/2). - Ilya Gutkovskiy, Nov 17 2021

A329444 The sixth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^6*binomial(n, m)^2.

Original entry on oeis.org

0, 1, 68, 1314, 18080, 197350, 1836792, 15233316, 115776768, 821760390, 5520171800, 35438827996, 219038609088, 1310833221724, 7629754810160, 43348888067400, 241117582878720, 1316197491501510, 7065439665315480, 37362065079691500, 194909773207512000, 1004374157379474420
Offset: 0

Views

Author

Nikita D. Gogin, Nov 16 2019

Keywords

References

  • H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Links

Crossrefs

Cf. A037966, A037972, A074334, A294486, A329521, A329913.

Programs

  • Magma
    [(&+[Binomial(n,k)^2*k^6: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 23 2022
  • Mathematica
    Table[Sum[m^6*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]
  • PARI
    a(n) = sum(m=0, n, m^6*binomial(n, m)^2); \\ Jinyuan Wang, Nov 23 2019
  • SageMath
    [n^3*(n+1)*(n^6+3*n^5-13*n^4-15*n^3+30*n^2+8*n-2)*catalan_number(n)/(8*(2*n-1)*(2*n-3)*(2*n-5)) for n in (0..30)] # G. C. Greubel, Jun 23 2022

Formula

a(n) = binomial(2*n, n) * n^3*(n^6 + 3*n^5 - 13*n^4 - 15*n^3 + 30*n^2 + 8*n - 2)/(8*(2*n-1)*(2*n-3)*(2*n-5)).
G.f.: x*(1 + 42*x - 168*x^2 + 1648*x^3 - 7608*x^4 + 18144*x^5 - 19376*x^6 - 1440*x^7 + 14400*x^8)/((1-4*x)^6*sqrt(1-4*x)). - G. C. Greubel, Jun 23 2022

A074334 a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.

Original entry on oeis.org

0, 1, 20, 234, 2144, 16750, 117432, 761460, 4654848, 27173718, 152867000, 834212236, 4438175040, 23108423884, 118111709744, 594059985000, 2946077521920, 14429322555750, 69892354873080, 335194270938780, 1593211647720000, 7511501237722020, 35153884344493200
Offset: 0

Views

Author

Paul Boddington, Mar 05 2003

Keywords

References

  • H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Links

Crossrefs

Cf. A000108 (Catalan numbers).
Cf. A000984, A002457, A037966, A037972, A329444, A329913.

Programs

  • Magma
    [n le 1 select n else n^2*(n^3+n^2-3*n-1)*Catalan(n-2): n in [0..30]]; // G. C. Greubel, Jun 23 2022
  • Mathematica
    Total/@Table[r^4 Binomial[n,r]^2,{n,0,20},{r,n}] (* Harvey P. Dale, Dec 04 2017 *)
Table[n^2*(n^3+n^2-3*n-1)*CatalanNumber[n-2] -Boole[n==1], {n,0,30}] (* G. C. Greubel, Jun 23 2022 *)
  • PARI
    vector(30, n, n--; sum(k=1, n, k^4*binomial(n,k)^2)) \\ Michel Marcus, Aug 19 2015
  • SageMath
    [n^2*(n^3+n^2-3*n-1)*catalan_number(n-2) for n in (0..30)] # G. C. Greubel, Jun 23 2022

Formula

For n>1 a(n) = n^2*(n^3+n^2-3*n-1)*C(n-2). Here C(n-2) = binomial(2*n-4, n-2)/(n-1) is a Catalan number.
From G. C. Greubel, Jun 23 2022: (Start)
a(n) = (n^2*(n^3 + n^2 - 3*n -1)/(2*(2*n-3)))*binomial(2*n-2, n-1).
G.f.: x*(1 + 2*x + 32*x^3 - 128*x^4 + 144*x^5)/(1-4*x)^(9/2).
E.g.f.: x*exp(2*x)*( (1+2*x)*(1 +6*x +4*x^2)*BesselI(0, 2*x) + 2*x*(2 + 7*x + 4*x^2)*BesselI(1, 2*x) ). (End)
D-finite with recurrence (n-1)*(39*n-106)*a(n) +4*(-38*n^2+n+290)*a(n-1) +4*(100*n^2-784*n+1145)*a(n-2) -64*(13*n+4)*(2*n-9)*a(n-3)=0. - R. J. Mathar, Sep 13 2024

Extensions

Terms a(18) and beyond from Andrew Howroyd, Jan 16 2020

A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

Original entry on oeis.org

1, 1, 8, 108, 2144, 56250, 1836792, 71799504, 3269445888, 169974711630, 9934458411800, 644825382429096, 46022332032100800, 3582265183110626740, 302002255041807372080, 27413749834141448520000, 2665789990569658618398720, 276477318687585566522176470
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 05 2020

Keywords

Links

Crossrefs

Cf. A000984, A002457, A037966, A037972, A072034, A074334, A187021, A329444, A329913, A336214, A341815.

Programs

  • Mathematica
    Join[{1}, Table[Sum[Binomial[n, k]^2 k^n, {k, 0, n}], {n, 1, 17}]]
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)^2*k^n); \\ Michel Marcus, Aug 05 2020

Formula

a(n) ~ c * d^n * (n-1)!, where d = (1 + 2*LambertW(exp(-1/2)/2)) / (4*LambertW(exp(-1/2)/2)^2) = 6.476217542109791521947605963458797355564... and c = 0.21617818094152997942246965143216887599763501682724844713834495... - Vaclav Kotesovec, Feb 20 2021

A336955 a(n) = Sum_{k=0..n} k^k * binomial(n, k)^2.

Original entry on oeis.org

1, 2, 9, 73, 849, 12651, 228493, 4836301, 117204545, 3196763983, 96842596701, 3224356269597, 116981406934417, 4591908332288837, 193846634326107701, 8755364023207809301, 421214258699748184321, 21500563181275847468503, 1160430732790051008442141, 66020998289431649938896445
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2020

Keywords

Crossrefs

Cf. A037966, A037972, A086331, A329913, A329444, A336828.

Programs

  • Mathematica
    a[n_] := 1 + Sum[k^k * Binomial[n, k]^2, {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Aug 09 2020 *)
  • PARI
    {a(n) = sum(k=0, n, k^k*binomial(n, k)^2)}
Showing 1-5 of 5 results.

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

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