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 11-17 of 17 results.

A337590 a(0) = 0; a(n) = n - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k) * k * a(k).

Original entry on oeis.org

0, 1, 0, -3, 28, -215, -174, 90223, -3840472, 103719537, 429704110, -357346077869, 35100093531900, -2005608652057595, -24108041118593418, 27881407632242902515, -4876442148527153942384, 474102062424164433715937, 12637408141631813073125094, -18867461801192524662360616421
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

Crossrefs

Cf. A002190, A009306, A336227.

Programs

  • Mathematica
    a[0] = 0; a[n_] := a[n] = n - (1/n) Sum[Binomial[n, k]^2 (n - k) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Log[1 + Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

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

A337824 a(0) = 0; a(n) = n^2 - (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (n-k))^2 * k * a(k).

Original entry on oeis.org

0, 1, 2, -15, 16, 2505, -60264, -606515, 131316928, -4813100271, -339213768200, 62401665573621, -2075963863814928, -745086903175541927, 140250562903680456332, 808225064553580739325, -5491409141464496462591744, 1013058261721909845376508449, 127689148764914765889971316600
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 24 2020

Keywords

Links

Crossrefs

Cf. A002190, A033464, A101981, A337590, A337591, A337825.

Programs

  • Maple
    S:= series(log(1+x*BesselI(0,2*sqrt(x))),x,31):
0,seq(coeff(S,x,n)*(n!)^2, n=1..30); # Robert Israel, Jan 07 2024
  • Mathematica
    a[0] = 0; a[n_] := a[n] = n^2 - (1/n) * Sum[(Binomial[n, k] (n - k))^2 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Log[1 + x BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

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

A337825 a(0) = 0; a(n) = n^3 - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k)^3 * k * a(k).

Original entry on oeis.org

0, 1, 6, -33, -512, 19405, 181116, -45817541, 771776384, 280415588121, -23151651942500, -3217963989270569, 816268626535923936, 38087192839910816485, -43268389662374707851552, 2822720920753640236252875, 3297662826737476255127428096, -833876355494162903256716734927
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 24 2020

Keywords

Crossrefs

Cf. A002190, A101981, A300452, A337590, A337824, A337826.

Programs

  • Mathematica
    a[0] = 0; a[n_] := a[n] = n^3 - (1/n) * Sum[Binomial[n, k]^2 (n - k)^3 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Log[1 + x (BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + x * (BesselI(0,2*sqrt(x)) + sqrt(x) * BesselI(1,2*sqrt(x)))).
Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + Sum_{n>=1} n^3 * x^n / (n!)^2).

A217940 Triangle read by rows: coefficients of polynomials Q_n(x) arising in study of Riemann zeta function.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 36, 33, 42, 33, 576, 480, 648, 720, 456, 14400, 10960, 14900, 18780, 17900, 9460, 518400, 362880, 487200, 648240, 730800, 606480, 274800, 25401600, 16465680, 21656040, 29481585, 36149820, 36569190, 26845140, 10643745, 1625702400, 981872640, 1260878080, 1729096320, 2218287120, 2495765440, 2285697120, 1503969600, 530052880
Offset: 1

Views

Author

N. J. A. Sloane, Oct 23 2012

Keywords

Examples

			Triangle begins:
1
1, 1
4, 4, 4
36, 33, 42, 33
576, 480, 648, 720, 456
14400, 10960, 14900, 18780, 17900, 9460
518400, 362880, 487200, 648240, 730800, 606480, 274800
...

Links

Crossrefs

Right-hand diagonal is A002190.

Programs

  • Mathematica
    Clear[q]; q[n_, 1] := (n-1)!^2; q[n_, k_] := q[n, k] = Sum[ Binomial[n-1, j]*Binomial[n-1, j+1]* Sum[q[j+1, r]*q[n-j-1, k-r], {r, Max[1, -n+j+k+1], Min[j+1, k-1]}], {j, 0, n-2}]; Table[q[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 13 2013 *)

Extensions

More terms from Jean-François Alcover, Feb 13 2013

A268482 Triangle that arise in the study of Galois polynomials.

Original entry on oeis.org

1, -1, 8, 4, -76, 264, -33, 1248, -9735, 22080, 456, -32088, 440448, -2085096, 3715440, -9460, 1216600, -26297700, 205444800, -704121000, 1087450320, 274800, -64995600, 2073673920, -23974142160, 129203087760, -354403429920, 500558083200
Offset: 1

Views

Author

Michel Marcus, Feb 05 2016

Keywords

Examples

			First few rows are:
1;
-1, 8;
4, -76, 264;
-33, 1248, -9735, 22080;
456, -32088, 440448, -2085096, 3715440;
...

Links

Crossrefs

Cf. A008292 (Eulerian numbers), A002190 (first column unsigned).

Programs

  • Mathematica
    c[k_] := c[k] = 1 - Sum[Binomial[k, j] Binomial[k-1, j-1] c[j], {j, k-1}];
eul[n_, x_] := Sum[(-1)^j Binomial[n+1, j] (x-j+1)^n, {j, 0, x+1}];
G[k_, m_] := G[k, m] = If [k==0 && m==0, 1, Sum[Binomial[k, j] Binomial[ k-1, j-1] c[j] Sum[eul[2j-1, i-1] G[k-j, m-i], {i, m}]/(2j-1)!, {j, k}]];
Table[(2n-1)! G[n, k], {n, 7}, {k, n}] // Flatten (* Jean-François Alcover, Sep 27 2018, from PARI *)
  • PARI
    C(k) = {my(j); 1 - sum(j=1, k-1, binomial(k, j)*binomial(k-1, j-1)*C(j))};
eul(n, x) = {my(j); sum(j=0, x+1, (-1)^j*binomial(n+1, j)*(x+1-j)^n)};
G(k, m) = if ((k==0) && (m==0), 1, sum(j=1, k, binomial(k,j)*binomial(k-1,j-1)*C(j)*sum(i=1, m, eul(2*j-1,i-1)*G(k-j, m-i))/(2*j-1)!));
tabl(nn) = for (n=1, nn, for (k=1, n, print1((2*n-1)!*G(n,k), ", "));print(););

A336609 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(1 / BesselJ(0,2*sqrt(x)) - 1).

Original entry on oeis.org

1, 1, 5, 52, 917, 24396, 909002, 45062697, 2862532213, 226403027044, 21794813189810, 2507115921526437, 339421509956163362, 53393907140415300317, 9653668439939308357991, 1987242385193691443059527, 461955240782446199029195253
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 27 2020

Keywords

Crossrefs

Cf. A000275, A002190, A023998.

Programs

  • Mathematica
    nmax = 16; CoefficientList[Series[Exp[1/BesselJ[0, 2 Sqrt[x]] - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
A000275[0] = 1; A000275[n_] := A000275[n] = -Sum[(-1)^(n - k) Binomial[n, k]^2 A000275[k], {k, 0, n - 1}]; a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k A000275[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * A000275(k) * a(n-k).

A372513 Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(BesselJ(0,2*sqrt(2*x))) / 2.

Original entry on oeis.org

0, 1, 2, 16, 264, 7296, 302720, 17587200, 1362399360, 135693537280, 16893684928512, 2570631845806080, 469393033744588800, 101294080603625226240, 25502237392032633323520, 7408331513180811911233536, 2459543337577081650719784960, 925435622656059412145504256000
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2024

Keywords

Crossrefs

Cf. A002190.

Programs

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

Formula

a(0) = 0; a(n) = (-2)^(n-1) - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (-2)^k * (n-k) * a(n-k).
a(n) = 2^(n-1) * A002190(n).
Previous Showing 11-17 of 17 results.

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

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