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 1-10 of 12 results. Next

A377529 Expansion of e.g.f. 1/(1 - x * exp(x))^2.

Original entry on oeis.org

1, 2, 10, 66, 560, 5770, 69852, 970886, 15228880, 266006610, 5119447700, 107617719022, 2453167135608, 60268223308826, 1587381621990556, 44619277892537910, 1333135910963656352, 42189279001183102882, 1409741875877923927332, 49597905017847180008126
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A006153, A377530.
Cf. A377503, A377527.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-x Exp[x])^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 04 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)*k^(n-k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1) * k^(n-k)/(n-k)!.
a(n) ~ n! * n/((1 + LambertW(1))^2 * LambertW(1)^n). - Vaclav Kotesovec, Oct 31 2024

A381207 Expansion of e.g.f. 1/(1 - x*cosh(x))^3.

Original entry on oeis.org

1, 3, 12, 69, 504, 4335, 43200, 490161, 6220032, 87242427, 1340305920, 22375475133, 403237638144, 7801208775399, 161245892161536, 3545854432602345, 82653484859228160, 2035605515838402291, 52814589875313573888, 1439814136866851346357, 41145786213980645621760
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A377530, A381209, A381210, A381211.
Cf. A205571, A381206.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (k+2)!*a185951(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * A185951(n,k).

A379942 Expansion of e.g.f. 1/( exp(-x) - x )^3.

Original entry on oeis.org

1, 6, 45, 411, 4449, 55803, 796581, 12757503, 226588257, 4420898595, 94001021589, 2163619250895, 53598352999905, 1421924243354787, 40221778417553637, 1208471542554184767, 38434396264371831873, 1289995362325669726659, 45567027291743788320405
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A072597, A379933, A379943.
Cf. A377530.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x)^3))
  • PARI
    a(n) = n!*sum(k=0, n, (k+3)^(n-k)*binomial(k+2, 2)/(n-k)!);

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A072597.
a(n) = n! * Sum_{k=0..n} (k+3)^(n-k) * binomial(k+2,2)/(n-k)!.

A379993 Expansion of e.g.f. 1/(1 - x * exp(x))^4.

Original entry on oeis.org

1, 4, 28, 252, 2776, 35940, 533304, 8908228, 165247072, 3368072196, 74782987240, 1796037420804, 46379441090448, 1281203788073092, 37694510810334616, 1176606639075726660, 38833052393329645504, 1351066066253778043908, 49417629820950190273992
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A379943, A379994, A379995, A379996.
Cf. A006153, A377529, A377530.

Programs

  • Mathematica
    nmax=18;CoefficientList[Series[1/(1 - x * Exp[x])^4,{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 05 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(k+3, 3)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(k+3,3)/(n-k)!.
a(n) == 0 (mod 4) for n>0.

A381209 Expansion of e.g.f. 1/(1 - x*cos(x))^3.

Original entry on oeis.org

1, 3, 12, 51, 216, 735, 0, -39081, -575232, -6047973, -48314880, -189159333, 3046957056, 99745485879, 1789140627456, 23433663134655, 185580069027840, -1250544374605389, -94781673979379712, -2543434372808424957, -47763303489939701760, -586864592847636893937
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A377530, A381207, A381210, A381211.
Cf. A352252, A381208.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (k+2)!*I^(n-k)*a185951(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381210 Expansion of e.g.f. 1/(1 - sinh(x))^3.

Original entry on oeis.org

1, 3, 12, 63, 408, 3123, 27552, 275103, 3065088, 37682883, 506606592, 7392091743, 116329479168, 1963781841843, 35395627487232, 678401549017983, 13776623985819648, 295481239628640003, 6674320861079273472, 158364407589097613823, 3937958237874411798528
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Crossrefs

Cf. A377530, A381207, A381209, A381211.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (k+2)!*a136630(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * A136630(n,k).

A381211 Expansion of e.g.f. 1/(1 - sin(x))^3.

Original entry on oeis.org

1, 3, 12, 57, 312, 1923, 13152, 98697, 805632, 7102563, 67233792, 679970937, 7315786752, 83421156003, 1004860895232, 12749105088777, 169926064668672, 2373678328434243, 34676591077097472, 528758667342524217, 8400613520498491392, 138830752520282729283
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Crossrefs

Cf. A377530, A381207, A381209, A381210.
Cf. A136630, A185690.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (k+2)!*I^(n-k)*a136630(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A380841 Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 21, 10, 3, 1, 0, 148, 66, 18, 4, 1, 0, 1305, 560, 141, 28, 5, 1, 0, 13806, 5770, 1380, 252, 40, 6, 1, 0, 170401, 69852, 16095, 2776, 405, 54, 7, 1, 0, 2403640, 970886, 217458, 35940, 4940, 606, 70, 8, 1, 0, 38143377, 15228880, 3335745, 533304, 70045, 8088, 861, 88, 9, 1
Offset: 0

Views

Author

Stefano Spezia, Feb 05 2025

Keywords

Examples

			Array begins as:
  1,    1,    1,     1,     1,     1,      1, ...
  0,    1,    2,     3,     4,     5,      6, ...
  0,    4,   10,    18,    28,    40,     54, ...
  0,   21,   66,   141,   252,   405,    606, ...
  0,  148,  560,  1380,  2776,  4940,   8088, ...
  0, 1305, 5770, 16095, 35940, 70045, 124350, ...
  ...

Crossrefs

Cf. A380843 (antidiagonal sums).
Columns k=0..4 give A000007, A006153, A377529, A377530, A379993.
Rows n=0..2 give A000012, A001477, A028552.
Main diagonal gives A380842.
A(n,n+1) gives A213643(n+1).

Programs

  • Mathematica
    A[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

Formula

A(n,k) = n! * Sum_{j=0..n} j^(n-j) * binomial(j+k-1,j)/(n-j)!. - Seiichi Manyama, Feb 06 2025

A377582 Expansion of e.g.f. (1 + x * exp(x))^3.

Original entry on oeis.org

1, 3, 12, 51, 228, 1035, 4698, 21063, 92424, 395091, 1643790, 6664383, 26387100, 102286587, 389125506, 1455994935, 5368721808, 19541252259, 70312410774, 250408115823, 883617559140, 3092276105163, 10740749281482, 37053754521831, 127037475064728, 433073722098675
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A002999, A377583.
Cf. A001815, A052791.
Cf. A377530, A377575.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3,k)/(n-k)!.
G.f.: (1-17*x+127*x^2-542*x^3+1453*x^4-2543*x^5+2973*x^6-2478*x^7+1626*x^8-648*x^9) / ((1-x)^2*(1-2*x)^3*(1-3*x)^4).

A379990 Expansion of e.g.f. exp(-2*x)/(exp(-x) - x)^3.

Original entry on oeis.org

1, 4, 25, 205, 2065, 24601, 337837, 5249581, 91006657, 1740663937, 36402220141, 826159146253, 20220201899377, 530828186303377, 14878044338021677, 443397290411503021, 14000282854007503105, 466866129420834410881, 16395362179348570608205, 604794784980600986425645
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A377530, A379942, A379991.
Cf. A092148, A377529.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(k+2, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * binomial(k+2,2)/(n-k)!.
a(n) ~ n! * n^2 / (2 * (LambertW(1) + 1)^3 * LambertW(1)^(n+1)). - Vaclav Kotesovec, Jan 08 2025
Showing 1-10 of 12 results. Next

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