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.

A186925 Coefficient of x^n in (1+n*x+x^2)^n.

Original entry on oeis.org

1, 1, 6, 45, 454, 5775, 88796, 1602447, 33213510, 777665691, 20302315252, 584774029983, 18422140045596, 630132567760345, 23257790717110392, 921362075184792825, 38994274473840538182, 1755943506127367745795, 83829045032101462204100, 4229207755493569286374167
Offset: 0

Views

Author

Emanuele Munarini, Mar 02 2011

Keywords

Links

Crossrefs

Main diagonal of A292627.
Cf. A092366, A187018, A187019, A187021, A070910, A292629.

Programs

  • Magma
    P:=PolynomialRing(Integers()); [ Coefficients((1+n*x+x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 02 2011
  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^(n-2*k), {k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[GegenbauerC[n, -n, -n/2] + KroneckerDelta[n, 0], {n, 0, 100}] (* Emanuele Munarini, Oct 20 2016 *)
Table[SeriesCoefficient[(1 + n*x + x^2)^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 13 2023 *)
  • Maxima
    a(n):=coeff(expand((1+n*x+x^2)^n),x,n);
  • Maxima
    makelist(ultraspherical(n,-n,-n/2),n,0,12); /* Emanuele Munarini, Oct 20 2016 */
makelist(a(n),n,0,20);
  • PARI
    {a(n) = sum(k=0, n, (n-2)^(n-k)*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, May 01 2019
  • PARI
    a(n) = polcoef((1+n*x+x^2)^n, n); \\ Michel Marcus, May 01 2019

Formula

a(n) = [x^n] (1+n*x+x^2)^n.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^(n-2*k).
a(n) ~ BesselI(0,2) * n^n. - Vaclav Kotesovec, Apr 17 2014
a(n) = GegenbauerPoly(n,-n,-n/2). - Emanuele Munarini, Oct 20 2016
From Ilya Gutkovskiy, Sep 20 2017: (Start)
a(n) = [x^n] 1/sqrt((1 + 2*x - n*x)*(1 - 2*x - n*x)).
a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*x). (End)
From Seiichi Manyama, May 01 2019: (Start)
a(n) = Sum_{k=0..n} (n-2)^(n-k) * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..n} (n+2)^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). (End)
a(n) = (1/4)^n * Sum_{k=0..n} (n-2)^k * (n+2)^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A187021 Coefficient of x^n in (1 + (n+1)*x + n*x^2)^n.

Original entry on oeis.org

1, 2, 13, 136, 1921, 33876, 712909, 17383584, 481003009, 14869654300, 507406003501, 18928740714192, 765897591633409, 33392080668673832, 1559976990077534253, 77717020110946293376, 4111810085670587224065, 230190619432401207833004, 13591965974806603671569101
Offset: 0

Views

Author

Emanuele Munarini, Mar 02 2011

Keywords

Links

Crossrefs

Main diagonal of A307883.
Cf. A092366, A186925, A187018, A187019, A241247, A234971.

Programs

  • Magma
    P:=PolynomialRing(Integers()); [ Coefficients((1+(n+1)*x+n*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011
  • Maple
    A187021:= n -> simplify( n^(n/2)*GegenbauerC(n, -n, -(n+1)/(2*sqrt(n))) );
1, seq(A187021(n), n = 1..30); # G. C. Greubel, May 31 2020
a := n -> hypergeom([-n, -n], [1], n):
seq(simplify(a(n)), n=0..18); # Peter Luschny, Dec 22 2020
  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n,k]^2*n^k,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[If[n==0, 1, Simplify[n^(n/2)*GegenbauerC[n, -n, -(n+1)/(2 Sqrt[n])]]], {n, 0, 30}] (* Emanuele Munarini, Oct 20 2016 *)
  • Maxima
    a(n):=coeff(expand((1+(n+1)*x+n*x^2)^n),x,n);
makelist(a(n),n,0,20);
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)^2*n^k)} \\ Paul D. Hanna, Mar 29 2011
  • Sage
    [1]+[ n^(n/2)*gegenbauer(n, -n, -(n+1)/(2*sqrt(n))) for n in (1..30)] # G. C. Greubel, May 31 2020

Formula

a(n) = [x^n] (1 + (n+1)*x + n*x^2)^n.
a(n) = n^(n/2)*GegenbauerPoly(n,-n,-(n+1)/(2*sqrt(n))). - Emanuele Munarini, Oct 20 2016
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k. - Paul D. Hanna, Mar 29 2011
a(n) ~ n^(n-1/4) * exp(2*sqrt(n)-1) / (2*sqrt(Pi)). - Vaclav Kotesovec, Apr 17 2014
a(n) = n! * [x^n] exp((n + 1)*x) * BesselI(0,2*sqrt(n)*x). - Ilya Gutkovskiy, May 31 2020
a(n) = hypergeom([-n, -n], [1], n). - Peter Luschny, Dec 22 2020

A187018 Coefficient of x^n in (1 + x + n*x^2)^n.

Original entry on oeis.org

1, 1, 5, 19, 145, 851, 7741, 58605, 600769, 5420035, 61026901, 628076153, 7648488145, 87388647373, 1138801242125, 14182492489651, 196218339243777, 2628971539313875, 38377805385510181, 547815690902283225, 8395817775835635601, 126725586542235932329
Offset: 0

Views

Author

Emanuele Munarini, Mar 02 2011

Keywords

Examples

			G.f. = 1 + x + 5*x^2 + 19*x^3 + 145*x^4 + 851*x^5 + 7741*x^6 + 58605*x^7 + ...

Links

Crossrefs

Cf. A092366, A186925, A187019, A187021.

Programs

  • Magma
    P:=PolynomialRing(Integers()); [ Coefficients((1+x+n*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011
  • Maple
    A187018:= n -> simplify( n^(n/2)*GegenbauerC(n, -n, -1/(2*sqrt(n))) );
1, seq(A187018(n), n = 1..30); # G. C. Greubel, May 31 2020
  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^k, {k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
a[ n_]:= SeriesCoefficient[ (1 + x + n*x^2)^n, {x, 0, n}]; (* Michael Somos, Dec 12 2014 *)
Table[If[n == 0, 1, Simplify[n^(n/2) GegenbauerC[n, -n, -1/(2 Sqrt[n])]]], {n, 0, 12}] (* Emanuele Munarini, Oct 20 2016 *)
  • Maxima
    a(n):=coeff(expand((1+x+n*x^2)^n),x,n);
makelist(a(n),n,0,20);
  • PARI
    {a(n)=polcoeff(1/sqrt(1 - 2*x - (4*n-1)*x^2 +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Dec 12 2014
  • PARI
    {a(n) = polcoef((1+x+n*x^2)^n, n)} \\ Seiichi Manyama, May 01 2019
  • Sage
    [1]+[ n^(n/2)*gegenbauer(n, -n, -1/(2*sqrt(n))) for n in (1..30)] # G. C. Greubel, May 31 2020

Formula

a(n) = [x^n] (1 + x + n*x^2)^n.
a(n) = n^(n/2)*GegenbauerPoly(n,-n,-1/(2*sqrt(n))). - Emanuele Munarini, Oct 20 2016
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^k.
a(n) ~ 2^(n-1/2) * exp(sqrt(n)/2-1/8) * n^(n/2-1/2) / sqrt(Pi). - Vaclav Kotesovec, Apr 17 2014
a(n) = [x^n] 1/sqrt(1 - 2*x - (4*n-1)*x^2). - Paul D. Hanna, Dec 12 2014
a(n) = n! * [x^n] exp(x) * BesselI(0,2*sqrt(n)*x). - Ilya Gutkovskiy, May 31 2020

A099169 a(n) = (1/n) * Sum_{k=0..n-1} C(n,k) * C(n,k+1) * (n-1)^k.

Original entry on oeis.org

1, 2, 11, 100, 1257, 20076, 387739, 8766248, 226739489, 6595646860, 212944033051, 7550600079672, 291527929539433, 12169325847587832, 545918747361417291, 26183626498897556176, 1336713063706757646465
Offset: 1

Views

Author

Ralf Stephan, Oct 09 2004

Keywords

Comments

A diagonal of Narayana array (A008550).

Links

Crossrefs

Cf. A092366, A187018, A187019, A187021.
Cf. A008550, A204057, A243631.

Programs

  • Magma
    A099169:= func< n | (&+[Binomial(n, j)*Binomial(n-1,j)*(n-1)^j/(j+1): j in [0..n-1]]) >;
[A099169(n): n in [1..30]]; // G. C. Greubel, Feb 16 2021
  • Maple
    A099169:= n-> add( binomial(n, j)*binomial(n-1,j)*(n-1)^j/(j+1), j=0..n-1);
seq( A099169(n), n=1..30) # G. C. Greubel, Feb 16 2021
  • Mathematica
    Join[{1},Table[Sum[Binomial[n,k]Binomial[n,k+1](n-1)^k,{k,0,n-1}]/n,{n,2,20}]] (* Harvey P. Dale, Oct 07 2013 *)
Table[Hypergeometric2F1[1-n,-n,2,-1+n],{n,1,20}] (* Vaclav Kotesovec, Apr 18 2014 *)
  • PARI
    a(n) = (1/n) * sum(k=0, n-1, binomial(n,k) * binomial(n,k+1) * (n-1)^k); \\ Michel Marcus, Feb 16 2021
  • Sage
    def A099169(n): return sum( binomial(n, j)*binomial(n-1,j)*(n-1)^j/(j+1) for j in [0..n-1])
[A099169(n) for n in [1..30]] # G. C. Greubel, Feb 16 2021

Formula

From Vaclav Kotesovec, Apr 18 2014, extended Dec 01 2021: (Start)
a(n) = Hypergeometric2F1([1-n,-n], [2], -1+n).
a(n) ~ exp(2*sqrt(n)-2) * n^(n-7/4) / (2*sqrt(Pi)) * (1 + 119/(48*sqrt(n))). (End)
Showing 1-4 of 4 results.

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