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.

Previous Showing 21-24 of 24 results.

A294642 a(n) = n! * [x^n] exp(n*x)*BesselI(1,2*sqrt(2)*x)/(sqrt(2)*x).

Original entry on oeis.org

1, 1, 6, 45, 456, 5825, 89896, 1627437, 33822944, 793783233, 20765009344, 599157626925, 18904594000128, 647524807918209, 23929038677825152, 948995910652193325, 40203601321988822528, 1812025020244371552897, 86577002960871477916672, 4371100278517527047687213
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2017

Keywords

Links

Crossrefs

Cf. A000108, A001003, A025235, A068764, A071356, A151374, A247496, A292716.

Programs

  • Mathematica
    Simplify[Table[n! SeriesCoefficient[Exp[n x] BesselI[1, 2 Sqrt[2] x]/(Sqrt[2] x), {x, 0, n}], {n, 0, 19}]]
Table[SeriesCoefficient[(1 - n x - Sqrt[1 - 2 n x + (n^2 - 8) x^2])/(4 x^2), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-2 x^2, 1 - n x, {i, 1, n}]), {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[2^k n^(n - 2 k) Binomial[n, 2 k] CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 1, 19}]]
Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {2}, 8/n^2], {n, 1, 19}]]

Formula

a(n) = [x^n] (1 - n*x - sqrt(1 - 2*n*x + (n^2 - 8)*x^2))/(4*x^2).
a(n) = [x^n] 1/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - ...))))), a continued fraction.
a(n) = Sum_{k=0..floor(n/2)} 2^k*n^(n-2*k)*binomial(n,2*k)*A000108(k).
a(n) = n^n*2F1(1/2-n/2,-n/2; 2; 8/n^2).
a(n) ~ c * n^n, where c = BesselI(1, 2*sqrt(2))/sqrt(2) = 2.3948330992734... - Vaclav Kotesovec, Nov 06 2017

A369213 Expansion of (1/x) * Series_Reversion( x / ((1+x)^4+x^2) ).

Original entry on oeis.org

1, 4, 23, 152, 1091, 8264, 65021, 526236, 4352942, 36637576, 312763225, 2701521420, 23567184019, 207343098824, 1837623853627, 16391011930424, 147029997389386, 1325506554640872, 12003342144724338, 109136630802023808, 995907341988015935
Offset: 0

Views

Author

Seiichi Manyama, Jan 16 2024

Keywords

Links

Crossrefs

Cf. A349331, A369126.
Cf. A001006, A065065, A071356.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^4+x^2))/x)
  • PARI
    a(n) = sum(k=0, n\2, binomial(n+1, k)*binomial(4*n-4*k+4, n-2*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(4*n-4*k+4,n-2*k).
D-finite with recurrence -3*(4813*n-632)*(3*n+2)*(3*n+4)*(n+1)*a(n) +2*(206141*n^4+1346849*n^3+118471*n^2-121301*n-7584)*a(n-1) +4*(1658281*n^4-3845638*n^3+4346111*n^2-2458136*n+406104)*a(n-2) +8*(n-2)*(2032705*n^3-6230304*n^2+5971619*n-935490)*a(n-3) +16*(n-2)*(n-3)*(958321*n^2-2152552*n+309963)*a(n-4) +544*(8765*n-1142)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 28 2024

A108073 Triangle in A071943 with rows reversed.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 9, 7, 3, 1, 31, 24, 12, 4, 1, 113, 89, 46, 18, 5, 1, 431, 342, 183, 76, 25, 6, 1, 1697, 1355, 741, 323, 115, 33, 7, 1, 6847, 5492, 3054, 1376, 520, 164, 42, 8, 1, 28161, 22669, 12768, 5900, 2326, 786, 224, 52, 9, 1, 117631, 94962, 54033, 25464
Offset: 0

Views

Author

N. J. A. Sloane, Jun 05 2005

Keywords

Comments

A convolution triangle based on A052709 (with first term omitted). - Philippe Deléham, Sep 15 2005

Examples

			1; 1,1; 3,2,1; 9,7,3,1; 31,24,12,4,1; ...

Crossrefs

Row sums yield A071356. Column 0 yields A052709.

Programs

  • Maple
    q:=sqrt(1-4*z-4*z^2): G:=(1-q)/z/(2-t+2*z+t*q): Gserz:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gserz,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form # Emeric Deutsch, Jun 06 2005
  • Mathematica
    T[n_, n_] = 1; T[n_, k_] := (k+1)*Sum[Binomial[i, n-k-i] * Binomial[k+2*i, i] / (k+i+1), {i, 1, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Vladimir Kruchinin *)
  • Maxima
    T(n,k):=if n=k then 1 else k*sum((binomial(i,n-k-i)*binomial(k+2*i-1,i))/(k+i),i,1,n-k); /* Vladimir Kruchinin, Apr 27 2015 */

Formula

G.f.: (1-q)/(z(2 - t + 2z + tq)), where q = sqrt(1 - 4z - 4z^2). - Emeric Deutsch, Jun 06 2005
T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n; T(n, k) = Sum_{j>=0} T(n-1, k-1+j) + Sum_{j>=0} T(n-1, k+1+j). - Philippe Deléham, Sep 15 2005
T(n,k) = k*Sum_{i=1..(n-k)} C(i,n-k-i)*C(k+2*i-1,i)/(k+i), n > k, T(n,n)=1. - Vladimir Kruchinin, Apr 27 2015

Extensions

More terms from Emeric Deutsch, Jun 06 2005

A177896 A binomial conjugate of the Narayana numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 9, 6, 1, 9, 26, 26, 10, 1, 21, 75, 100, 60, 15, 1, 51, 216, 360, 295, 120, 21, 1, 127, 623, 1246, 1295, 735, 217, 28, 1, 323, 1800, 4200, 5292, 3864, 1624, 364, 36, 1, 835, 5211, 13896, 20580, 18396, 10080, 3276, 576, 45, 1, 2188, 15115, 45345, 77190, 81690, 55314, 23730, 6150, 870, 55, 1
Offset: 0

Views

Author

Paul Barry, Dec 15 2010

Keywords

Comments

Equals P^{-1}*N*P where P is A007318 and N is A001263.
First column is Motzkin numbers A001006. Row sums are A071356.
Diagonal sums are A078481.

Examples

			Triangle begins
1,
1, 1,
2, 3, 1,
4, 9, 6, 1,
9, 26, 26, 10, 1,
21, 75, 100, 60, 15, 1,
51, 216, 360, 295, 120, 21, 1,
127, 623, 1246, 1295, 735, 217, 28, 1,
323, 1800, 4200, 5292, 3864, 1624, 364, 36, 1

Formula

G.f.: ( 1-x(1+y)-sqrt(1-2x(1+y)+x^2(1+y)(y-3)) )/(2x^2(1+y));
G.f.: 1/(1-x-xy-x^2(1+y)/(1-x-xy-x^2(1+y)/(1-... (continued fraction).
Previous Showing 21-24 of 24 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.