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.

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A308946 Expansion of e.g.f. 1/(1 - x*(1 + x/2)*exp(x)).

Original entry on oeis.org

1, 1, 5, 30, 244, 2485, 30351, 432502, 7043660, 129050649, 2627117875, 58829021416, 1437117395946, 38032508860177, 1083932872119839, 33098858988564090, 1078083456543449416, 37309607437056658129, 1367138649165397662627, 52879280631976735387588
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 02 2019

Keywords

Crossrefs

Cf. A000217, A001793, A006153, A052529, A279361, A308861.

Programs

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

Formula

E.g.f.: 1 / (1 - Sum_{k>=1} (k*(k + 1)/2)*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000217(k) * a(n-k).
a(n) ~ n! * (2 + r) / ((2 + 4*r + r^2) * r^n), where r = 0.49122518354447387971550543450091640839121607... is the root of the equation exp(r)*r*(2 + r) = 2. - Vaclav Kotesovec, Aug 09 2021

A340904 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_1(k) * a(n-k).

Original entry on oeis.org

1, 1, 5, 28, 225, 2206, 26174, 361278, 5704401, 101297701, 1998893240, 43386854622, 1027353587730, 26353742447280, 728030940612638, 21548668265211778, 680330296613877761, 22821706122361385354, 810587673640374442445, 30390159250481750866640
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 26 2021

Keywords

Links

Crossrefs

Cf. A000203, A006153, A180305, A274804, A340903.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] DivisorSigma[1, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(1 - Sum[Sum[i x^(i j)/(i j)!, {j, 1, nmax}], {i, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, sigma(k)*x^k/k!)))) \\ Seiichi Manyama, Mar 29 2022
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, sigma(k)*binomial(n, k)*a(n-k))); \\ Seiichi Manyama, Mar 29 2022

Formula

E.g.f.: 1 / (1 - Sum_{i>=1} Sum_{j>=1} i * x^(i*j) / (i*j)!).
E.g.f.: 1 / (1 - Sum_{k>=1} sigma_1(k) * x^k/k!). - Seiichi Manyama, Mar 29 2022

A351790 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (k * (n-j))^j/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 6, 21, 24, 1, 1, 8, 42, 148, 120, 1, 1, 10, 69, 392, 1305, 720, 1, 1, 12, 102, 780, 4600, 13806, 5040, 1, 1, 14, 141, 1336, 11145, 64752, 170401, 40320, 1, 1, 16, 186, 2084, 22200, 191178, 1063216, 2403640, 362880
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Examples

			Square array begins:
    1,    1,    1,     1,     1,     1, ...
    1,    1,    1,     1,     1,     1, ...
    2,    4,    6,     8,    10,    12, ...
    6,   21,   42,    69,   102,   141, ...
   24,  148,  392,   780,  1336,  2084, ...
  120, 1305, 4600, 11145, 22200, 39145, ...

Crossrefs

Columns k=0..4 give A000142, A006153, A336950, A336951, A336952.
Main diagonal gives A235328.
Cf. A351761, A351791.

Programs

  • Mathematica
    T[n_, k_] := n!*(1 + Sum[(k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)
  • PARI
    T(n, k) = n!*sum(j=0, n, (k*(n-j))^j/j!);
  • PARI
    T(n, k) = if(n==0, 1, n*sum(j=0, n-1, k^(n-1-j)*binomial(n-1, j)*T(j, k)));

Formula

E.g.f. of column k: 1/(1 - x*exp(k*x)).
T(0,k) = 1 and T(n,k) = n * Sum_{j=0..n-1} k^(n-1-j) * binomial(n-1,j) * T(j,k) for n > 0.

A377526 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^5.

Original entry on oeis.org

1, 1, 12, 273, 9604, 460105, 27966126, 2062219117, 178897527768, 17853102321489, 2014988044093210, 253792946798597701, 35290880970687039732, 5370055269772474994713, 887591963820839894529654, 158357028389450319651183165, 30332317748593431632078480176, 6208425034878692992471996557217
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Comments

In general, for k > 1, if e.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^k, then a(n) ~ sqrt(k*(1 + LambertW((k-1)^(k-1)/k^k))) * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW((k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Nov 11 2024

Crossrefs

Cf. A006153, A295238, A364983, A364987.
Cf. A002294.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k, k)/((4*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002294(k)/(n-k)!.
a(n) ~ sqrt(5*(1 + LambertW(256/3125))) * n^(n-1) / (8 * exp(n) * LambertW(256/3125)^n). - Vaclav Kotesovec, Nov 11 2024

A377575 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^3.

Original entry on oeis.org

1, 3, 30, 483, 11100, 334035, 12478698, 558058179, 29104042152, 1735547479587, 116539815603630, 8704631976941043, 716019297815418732, 64326542671867079955, 6267631435921525638738, 658359915933162131600355, 74168964857766293453918928, 8921104769819780822122624323
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A006153, A377574, A377576.
Cf. A364983, A377554.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*k+3, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364983.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+3,k)/( (k+1)*(n-k)! ).

A377576 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^4.

Original entry on oeis.org

1, 4, 52, 1116, 34408, 1394340, 70298424, 4248802516, 299752943200, 24196951718532, 2200519882434280, 222683725755611604, 24824104612186789584, 3023063956714780554628, 399343825987950226379416, 56879649386095684434783060, 8689968793295620150120679104
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A006153, A377574, A377575.
Cf. A364987, A377577.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k+4, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A364987.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+4,k)/( (k+1)*(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.

A380155 Expansion of e.g.f. 1/sqrt(1 - 2*x*exp(2*x)).

Original entry on oeis.org

1, 1, 7, 63, 785, 12545, 244407, 5619775, 148977313, 4473497601, 150078670055, 5563415292479, 225832882678449, 9962766560986369, 474619650950131351, 24283168467229957695, 1327993894505461755713, 77305844496338607597569, 4772660185400974888323015
Offset: 0

Views

Author

Seiichi Manyama, Jan 13 2025

Keywords

Crossrefs

Cf. A006153, A380156.
Cf. A380014, A380015.

Programs

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

Formula

a(n) = 2^n * n! * Sum_{k=0..n} (-1)^k * k^(n-k) * binomial(-1/2,k)/(n-k)!.
a(n) == 1 (mod 2).
a(n) ~ 2^(n + 1/2) * n^n / (sqrt(1 + LambertW(1)) * exp(n) * LambertW(1)^n). - Vaclav Kotesovec, Jan 23 2025

A305407 Expansion of e.g.f. 1/(1 + log(1 - x)*exp(x)).

Original entry on oeis.org

1, 1, 5, 32, 274, 2939, 37833, 568210, 9753280, 188342949, 4041170695, 95380234366, 2455830637412, 68501591450447, 2057726452045145, 66227424015265178, 2273614433910697920, 82932491842062712873, 3202994529476330549163, 130577628147690206429038, 5603479009890212632226756
Offset: 0

Views

Author

Ilya Gutkovskiy, May 31 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 32*x^3/3! + 274*x^4/4! + 2939*x^5/5! + 37833*x^6/6! + ...

Links

Crossrefs

Cf. A002104, A006153, A007840, A009324, A191365.

Programs

  • Maple
    a:=series(1/(1+log(1-x)*exp(x)),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x] Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[HypergeometricPFQ[{1, 1, 1 - k}, {2}, -1] a[n - k]/(k - 1)!, {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

Formula

a(n) ~ n! / ((1 + exp(r)/r) * (1 - exp(-r))^(n+1)), where r = 0.62747017959751658496114808922921433658821962606026068561095... is the root of the equation r*exp(1 - exp(-r)) = 1. - Vaclav Kotesovec, Mar 26 2019
a(0) = 1; a(n) = Sum_{k=1..n} A002104(k) * binomial(n,k) * a(n-k). - Seiichi Manyama, May 04 2022

A351768 a(n) = n! * Sum_{k=0..n} k^(n-k) * (n-k)^k/k!.

Original entry on oeis.org

1, 0, 2, 18, 276, 6260, 190950, 7523082, 371286440, 22356290952, 1608686057610, 136069954606190, 13345029902628732, 1500054487474871484, 191349476316804534638, 27464505325501082617170, 4402551348139824475260240, 783025812197886669354545552
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2022

Keywords

Crossrefs

Cf. A006153, A062817, A134095, A351765, A351780.

Programs

  • Mathematica
    Join[{1}, Table[n!*Sum[k^(n-k) * (n-k)^k/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Feb 19 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*(n-k)^k/k!);

Formula

log(a(n)) ~ n *(2*log(n) - log(log(n)) - 2 + (log(log(n)) + log(log(n)-1) + 1)/log(n)). - Vaclav Kotesovec, Feb 19 2022
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