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 1-7 of 7 results.

A264036 Stirling transform of A077957 (aerated powers of 2).

Original entry on oeis.org

1, 0, 2, 6, 18, 70, 330, 1694, 9202, 53334, 332090, 2212782, 15638370, 116365990, 907975146, 7413080510, 63212284498, 561747543414, 5190343710746, 49752410984526, 493844719701186, 5068209425457862, 53705511911500746, 586862875255860062, 6605213319604075186
Offset: 0

Views

Author

Vladimir Reshetnikov, Nov 01 2015

Keywords

Comments

a(n) is the inverse binomial transform of A264037 without the leading zero [1, 1, 3, 13, 55, ...].

Examples

			G.f. = 1 + 2*x^2 + 6*x^3 + 18*x^4 + 70*x^5 + 330*x^6 + 1694*x^7 + 9202*x^8 + ...

Links

Crossrefs

Column k=2 of A357681.
Cf. A065143, A077957, A264037.

Programs

  • Mathematica
    Table[(BellB[n, Sqrt[2]] + BellB[n, -Sqrt[2]])/2, {n, 0, 24}]
  • PARI
    vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k, 2))) \\ Altug Alkan, Nov 01 2015

Formula

a(n) = Sum_{k=0..n} A077957(k)*Stirling2(n,k).
a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k).
a(n) = (Bell_n(sqrt(2)) + Bell_n(-sqrt(2)))/2, where Bell_n(x) is n-th Bell polynomial.
Bell_n(sqrt(2)) = a(n) + A264037(n)*sqrt(2).
E.g.f.: cosh(sqrt(2)*(exp(x) - 1)).
a(n) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k) * A264037(k). - Seiichi Manyama, Oct 12 2022

A357598 Expansion of e.g.f. sinh(2 * (exp(x)-1)) / 2.

Original entry on oeis.org

0, 1, 1, 5, 25, 117, 601, 3509, 22457, 153141, 1105561, 8453557, 68339833, 581495605, 5184047961, 48259748533, 468040609593, 4719817792565, 49396003390489, 535526127566773, 6004124908829177, 69509047405180213, 829801009239621849, 10202835010223731893
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2022

Keywords

Links

Crossrefs

Cf. A024429, A264037, A357572.
Cf. A065143, A078944, A357599.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(2*(exp(x)-1))/2)))
  • PARI
    a(n) = sum(k=0, (n-1)\2, 4^k*stirling(n, 2*k+1, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 2)-Bell_poly(n, -2)))/4;

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(2) - Bell_n(-2) )/4, where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A065143(k).

A357572 Expansion of e.g.f. sinh(sqrt(3) * (exp(x)-1)) / sqrt(3).

Original entry on oeis.org

0, 1, 1, 4, 19, 85, 406, 2191, 13105, 84190, 573121, 4127521, 31434184, 252388957, 2126998693, 18740283556, 172134162631, 1644920020417, 16324076578870, 167938152551491, 1787952325142341, 19667748794844550, 223217829954224029, 2610546296216999197
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2022

Keywords

Links

Crossrefs

Cf. A024429, A264037, A357598.
Cf. A027710, A357615, A357737.

Programs

  • PARI
    a(n) = sum(k=0, (n-1)\2, 3^k*stirling(n, 2*k+1, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(3))-Bell_poly(n, -sqrt(3)))/(2*sqrt(3)));

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 3^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(sqrt(3)) - Bell_n(-sqrt(3)) )/(2 * sqrt(3)), where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357615(k).

A357664 Expansion of e.g.f. sinh( (exp(2*x) - 1)/sqrt(2) )/sqrt(2).

Original entry on oeis.org

0, 1, 2, 6, 32, 220, 1592, 11944, 96000, 847120, 8209952, 86020704, 958326272, 11243157952, 138464594816, 1789358629504, 24250275913728, 344002396594432, 5092763802452480, 78443316497892864, 1253887341918199808, 20761127890765634560
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Cf. A024429, A357665, A357666.
Cf. A009599, A264037, A357661.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sinh((exp(2*x)-1)/sqrt(2))/sqrt(2)))))
  • PARI
    a(n) = sum(k=0, (n-1)\2, 2^(n-1-k)*stirling(n, 2*k+1, 2));

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 2^(n-1-k) * Stirling2(n,2*k+1).

A357668 Expansion of e.g.f. sinh( 3 * (exp(x) - 1) )/3.

Original entry on oeis.org

0, 1, 1, 10, 55, 307, 2026, 14779, 114157, 933616, 8110261, 74525167, 719925328, 7279859485, 76855303981, 845280487018, 9663800287483, 114601481983855, 1407040763488354, 17856103120048783, 233883061849700137, 3157648445216335528, 43887908697233605489
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2022

Keywords

Links

Crossrefs

Cf. A024429, A264037, A357572, A357598.
Cf. A027710, A356572, A357667.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(3*(exp(x)-1))/3)))
  • PARI
    a(n) = sum(k=0, (n-1)\2, 9^k*stirling(n, 2*k+1, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 3)-Bell_poly(n, -3)))/6;

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(3) - Bell_n(-3) )/6, where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357667(k).

A323631 Stirling transform of Pell numbers (A000129).

Original entry on oeis.org

0, 1, 3, 12, 57, 305, 1798, 11531, 79707, 589426, 4634471, 38547861, 337734048, 3105588629, 29877483743, 299906019892, 3133423928557, 34002824654365, 382507638525838, 4452923233600903, 53561431659306039, 664728428775177890, 8500763141347126563, 111886109022440334593, 1513989730079050155936
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 21 2019

Keywords

Links

Crossrefs

Cf. A000110, A000129, A263575, A263576, A264037.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
      (<<2|1>, <1|0>>^m)[1, 2], m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jun 23 2023
  • Mathematica
    FullSimplify[nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] Sinh[Sqrt[2] (Exp[x] - 1)]/Sqrt[2], {x, 0, nmax}], x] Range[0, nmax]!]
Table[Sum[StirlingS2[n, k] Fibonacci[k, 2], {k, 0, n}], {n, 0, 24}]
Table[Sum[Binomial[n, k] BellB[n - k] (BellB[k, Sqrt[2]] - BellB[k, -Sqrt[2]])/(2 Sqrt[2]), {k, 0, n}], {n, 0, 24}]

Formula

E.g.f.: exp(exp(x) - 1)*sinh(sqrt(2)*(exp(x) - 1))/sqrt(2).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A000129(k).
a(n) = Sum_{k=0..n} binomial(n,k)*A000110(n-k)*A264037(k).

A357736 Expansion of e.g.f. sin( sqrt(2) * (exp(x) - 1) )/sqrt(2).

Original entry on oeis.org

0, 1, 1, -1, -11, -45, -119, -49, 2045, 18075, 105121, 436471, 679669, -10538333, -155858247, -1404609569, -9667430739, -46708291093, -25694453615, 3002522206471, 49051481154341, 546022210068595, 4800733688293929, 31399017314213487, 75507020603213405
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2022

Keywords

Links

Crossrefs

Cf. A264037, A357725.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sin(sqrt(2)*(exp(x)-1))/sqrt(2)))))
  • PARI
    a(n) = sum(k=0, (n-1)\2, (-2)^k*stirling(n, 2*k+1, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(2)*I)-Bell_poly(n, -sqrt(2)*I))/(2*sqrt(2)*I));

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} (-2)^(k) * Stirling2(n,2*k+1).
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357725(k).
a(n) = ( Bell_n(sqrt(2) * i) - Bell_n(-sqrt(2) * i) )/(2 * sqrt(2) * i), where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.
Showing 1-7 of 7 results.

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