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 11-20 of 20 results.

A340887 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 3^(n-k-1) * a(k).

Original entry on oeis.org

1, 1, 7, 99, 2511, 99531, 5680125, 441226521, 44766049599, 5748319130283, 911271895816077, 174799606363478361, 39903413238125862309, 10690643656077551475921, 3321750648705212259711063, 1184831658624977151885176859, 480843465699932167142334581919
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 25 2021

Keywords

Crossrefs

Cf. A102221, A255927, A340886, A340888.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 3^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[3/(4 - BesselI[0, 2 Sqrt[3 x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = 3 / (4 - BesselI(0,2*sqrt(3*x))).

A340888 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 4^(n-k-1) * a(k).

Original entry on oeis.org

1, 1, 8, 124, 3456, 150656, 9453056, 807373568, 90066059264, 12716049596416, 2216452086693888, 467465806422867968, 117332539562036035584, 34562989958399757647872, 11807922834511544081973248, 4630865359842075866336067584, 2066370767828213666946077425664
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 25 2021

Keywords

Crossrefs

Cf. A102221, A326324, A340886, A340887.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 4^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[4/(5 - BesselI[0, 4 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = 4 / (5 - BesselI(0,4*sqrt(x))).

A352467 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k)^2 * a(n-k).

Original entry on oeis.org

1, 1, 37, 8551, 6886069, 14323022551, 64085654997739, 545107167737695109, 8062740187879748199029, 193866963305030079530064391, 7188682292472952994057436691387, 394013888612808806428687953794890229, 30829606055995735731623164115609901072859
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2022

Keywords

Crossrefs

Cf. A094088, A102221, A352468, A352470.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k]^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 12}]
nmax = 24; Take[CoefficientList[Series[1/(1 - Sum[x^(2 k)/(2 k)!^2, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2, {1, -1, 2}]

Formula

Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = 1 / (1 - Sum_{n>=1} x^(2*n) / (2*n)!^2).
Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = 1 / (2 - (BesselI(0,2*sqrt(x)) + BesselJ(0,2*sqrt(x))) / 2).

A352470 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)^2 * a(n-2*k-1).

Original entry on oeis.org

1, 1, 4, 37, 608, 15601, 576472, 28993693, 1904637184, 158352856129, 16253786050904, 2018684970206653, 298373110433984192, 51757706826973479697, 10412613242348421164400, 2404755328388872932588037, 631887117002962512609921024, 187441600433239155105076467457
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2022

Keywords

Crossrefs

Cf. A006154, A102221, A346220, A352467, A352471.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k + 1]^2 a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)/(2 k + 1)!^2, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / n!^2 = 1 / (1 - Sum_{n>=0} x^(2*n+1) / (2*n+1)!^2).
Sum_{n>=0} a(n) * x^n / n!^2 = 1 / (1 - (BesselI(0,2*sqrt(x)) - BesselJ(0,2*sqrt(x))) / 2).

A335946 a(n) = 1 + Sum_{k=0..n-1} binomial(n,k)^2 * a(k).

Original entry on oeis.org

1, 2, 10, 110, 2154, 65902, 2903446, 174109546, 13636888810, 1351801926542, 165434393561910, 24497621303302666, 4317170011370444982, 892891315599103615082, 214174328063904077240962, 58974283594413521123672110, 18476316023495768160707616490
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 01 2020

Keywords

Links

Crossrefs

Row sums of A102220.
Cf. A000629, A102221.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(2 - BesselI[0, 2 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (2 - BesselI(0,2*sqrt(x))).
a(n) = 2 * A102221(n) for n > 0.

A336228 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).

Original entry on oeis.org

1, 1, 6, 75, 1684, 59005, 2977566, 204512875, 18346977608, 2083115635065, 291996210173410, 49525220811387871, 9996609976117991436, 2368117724291275331869, 650613686811158069472942, 205196311013650099853516115, 73633144885479474283911225616
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 12 2020

Keywords

Crossrefs

Cf. A006153, A102221.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - Sqrt[x] BesselI[1, 2 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

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

A336243 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * (n-k-1)! * a(k).

Original entry on oeis.org

1, 1, 5, 56, 1114, 34624, 1549648, 94402356, 7511324448, 756406501200, 94039208461584, 14146468841290752, 2532586289913605088, 532113978869395649856, 129662518122880634567232, 36270261084908437106586624, 11543682123659880166705099776
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 13 2020

Keywords

Crossrefs

Cf. A007840, A102221, A110083, A193161.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - Sum[x^k/(k k!), {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 16; Assuming[x > 0, CoefficientList[Series[1/(1 + EulerGamma - ExpIntegralEi[x] + Log[x]), {x, 0, nmax}], x]] Range[0, nmax]!^2

Formula

a(n) = (n!)^2 * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k*k!)).

A102224 Column 0 of the matrix square of A102220, which equals the lower triangular matrix: [2*I - A008459]^(-1).

Original entry on oeis.org

1, 2, 14, 200, 4814, 174752, 8909168, 606818060, 53211837134, 5838211285616, 783434682568664, 126221710572107900, 24043148814317769584, 5344827109234104188348, 1371307353540074156012828
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2004

Keywords

Comments

A102221 is column 0 of A102220.
Triangle A008459 consists of the squared binomial coefficients.

Examples

			Given A102221 = [1,1,5,55,1077,32951,1451723,87054773,...], then this sequence results from a type of self-convolution of A102221:
a(2) = 14 = 1^2*1*5 + 2^2*1*1 + 1^2*5*1,
a(3) = 200 = 1^2*1*55 + 3^2*1*5 + 3^2*5*1 + 1^2*55*1.

Crossrefs

Cf. A102220, A102221, A008459.

Programs

  • PARI
    {a(n)=(matrix(n+1,n+1,i,j,if(i==j,2,0)-binomial(i-1,j-1)^2)^-2)[n+1,1]}

Formula

a(n) = Sum_{k=0..n} C(n, k)^2*A102221(k)*A102221(n-k).
Sum_{n>=0} a(n)*x^n/n!^2 = 1/(2-BesselI(0,2*sqrt(x)))^2. - Vladeta Jovovic, Jul 17 2006

A336217 a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n,k)^2 * a(k).

Original entry on oeis.org

1, 2, 18, 362, 12946, 723402, 58208490, 6375093258, 911949196434, 165104835435146, 36903191037412618, 9980525774650881738, 3212329170232153022314, 1213419234370490738427722, 531582989226188067128503722, 267336170027296964096123899962
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 12 2020

Keywords

Crossrefs

Cf. A004123, A102221.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; CoefficientList[Series[1/(1 - 2 Sum[x^k/(k!)^2, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

a(n) = (n!)^2 * [x^n] 1 / (1 - 2 * Sum_{k>=1} x^k / (k!)^2).
a(n) = (n!)^2 * [x^n] 1 / (3 - 2 * BesselI(0,2*sqrt(x))).
a(n) ~ (n!)^2 / (2 * BesselI(1, 2*sqrt(r)) * r^(n + 1/2)), where r = 0.4473998881770456142157108538567782213913712561... is the root of the equation 2*BesselI(0, 2*sqrt(r)) = 3. - Vaclav Kotesovec, Jul 17 2020

A362589 Triangular array read by rows. T(n,k) is the number of ways to form an ordered pair of n-permutations and then choose a size k subset of its common descent set, n >= 0, 0 <= k <= max{0,n-1}.

Original entry on oeis.org

1, 1, 4, 1, 36, 18, 1, 576, 432, 68, 1, 14400, 14400, 3900, 250, 1, 518400, 648000, 252000, 32400, 922, 1, 25401600, 38102400, 19404000, 3880800, 262542, 3430, 1, 1625702400, 2844979200, 1795046400, 493920000, 56664384, 2119152, 12868, 1
Offset: 0

Views

Author

Geoffrey Critzer, May 01 2023

Keywords

Examples

			Triangle begins:
     1;
     1;
     4,     1;
    36,    18,    1;
   576,   432,   68,   1;
 14400, 14400, 3900, 250, 1;
 ...

Links

Crossrefs

Cf. A001044 (column k=0), A102221 (row sums), A192721.

Programs

  • Mathematica
    nn = 8; B[n_] := n!^2; e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[u/(u + 1 - e[u z]), {z, 0, nn}], {z, u}]] // Flatten

Formula

Sum_{n>=0} Sum_{k=0..n-1} T(n,k)*u^k*z^n/(n!)^2 = u/(u + 1 - E(u*z)) where E(z) = Sum_{n>=0} z^n/(n!)^2.
Column k=1: Sum_{k=1..n-1} A192721(n,k)*k gives total number of common descents over all permutation pairs.
Previous Showing 11-20 of 20 results.

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

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