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 1-6 of 6 results.

A357681 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * (exp(x) - 1) ).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 8, 0, 1, 0, 4, 9, 18, 25, 0, 1, 0, 5, 12, 30, 70, 97, 0, 1, 0, 6, 15, 44, 135, 330, 434, 0, 1, 0, 7, 18, 60, 220, 705, 1694, 2095, 0, 1, 0, 8, 21, 78, 325, 1228, 3906, 9202, 10707, 0, 1, 0, 9, 24, 98, 450, 1905, 7196, 22953, 53334, 58194, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2022

Keywords

Examples

			Square array begins:
  1,  1,  1,   1,   1,   1, ...
  0,  0,  0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5, ...
  0,  3,  6,   9,  12,  15, ...
  0,  8, 18,  30,  44,  60, ...
  0, 25, 70, 135, 220, 325, ...

Links

Crossrefs

Columns k=0-4 give: A000007, A024430, A264036, A357615, A065143.
Column k=9 gives A357667.
Main diagonal gives A357682.
Cf. A292860.

Programs

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

Formula

T(n,k) = Sum_{j=0..floor(n/2)} k^j * Stirling2(n,2*j).
T(n,k) = ( Bell_n(sqrt(k)) + Bell_n(-sqrt(k)) )/2, where Bell_n(x) is n-th Bell polynomial.

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).

A357726 Expansion of e.g.f. cos( sqrt(3) * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, -3, -9, -12, 45, 465, 2394, 7827, 639, -250410, -2588553, -17773635, -84525480, -105849399, 3569654115, 56100280308, 561682625769, 4227837863181, 20472943653306, -38990802816489, -2621206974761253, -42512769453705474, -495174030273565173
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=3 of A357728.
Cf. A357615, A357737.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Cos[Sqrt[3](Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 20 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(3)*(exp(x)-1)))))
  • PARI
    a(n) = sum(k=0, n\2, (-3)^k*stirling(n, 2*k, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(3)*I)+Bell_poly(n, -sqrt(3)*I)))/2;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-3)^k * Stirling2(n,2*k).
a(n) = 1; a(n) = -3 * Sum_{k=0..n-1} binomial(n-1, k) * A357737(k).
a(n) = ( Bell_n(sqrt(3) * i) + Bell_n(-sqrt(3) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

A357662 Expansion of e.g.f. cosh( (exp(3*x) - 1)/sqrt(3) ).

Original entry on oeis.org

1, 0, 3, 27, 198, 1485, 12825, 132678, 1582497, 20603727, 284290560, 4132840239, 63571690485, 1038868740000, 18022911716439, 330305863479615, 6355242571945878, 127721845479277737, 2672729031195365949, 58142565625982730462, 1313557910179640120061
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Cf. A024430, A357661, A357663.
Cf. A357615, A357649, A357665.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cosh((exp(3*x)-1)/sqrt(3)))))
  • PARI
    a(n) = sum(k=0, n\2, 3^(n-k)*stirling(n, 2*k, 2));

Formula

a(n) = Sum_{k=0..floor(n/2)} 3^(n-k) * Stirling2(n,2*k).

A357667 Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 9, 27, 144, 945, 6273, 44226, 339399, 2796795, 24387786, 223853355, 2159078445, 21827316888, 230536050165, 2536213188519, 28994911890048, 343806474384045, 4220933769308205, 53566838971016418, 701650841036287275, 9473067208871584407
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2022

Keywords

Links

Crossrefs

Column k=9 of A357681.
Cf. A024430, A065143, A264036, A357615.
Cf. A027710, A357649, A357668.

Programs

  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(cosh(3*(exp(x)-1))))
  • PARI
    a(n) = sum(k=0, n\2, 9^k*stirling(n, 2*k, 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)))/2;

Formula

E.g.f.: cosh( 3 * (exp(x) - 1) ).
a(n) = Sum_{k=0..floor(n/2)} 9^k * Stirling2(n,2*k).
a(n) = ( Bell_n(3) + Bell_n(-3) )/2, where Bell_n(x) is n-th Bell polynomial.
a(n) = 1; a(n) = 9 * Sum_{k=0..n-1} binomial(n-1, k) * A357668(k).

A357703 Expansion of e.g.f. cosh( sqrt(3) * log(1-x) ).

Original entry on oeis.org

1, 0, 3, 9, 42, 240, 1614, 12474, 108900, 1059696, 11371932, 133410420, 1698541416, 23324023008, 343606235544, 5405580540360, 90445832210448, 1603781918563968, 30042007763367600, 592788643008571152, 12289695299276133024, 267079782474700715520
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=3 of A357712.
Cf. A357615.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cosh(sqrt(3)*log(1-x)))))
  • PARI
    a(n) = sum(k=0, n\2, 3^k*abs(stirling(n, 2*k, 1)));
  • PARI
    a(n) = round((prod(k=0, n-1, sqrt(3)+k)+prod(k=0, n-1, -sqrt(3)+k)))/2;
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=2, n, v[i+1]=(2*i-3)*v[i]-(i^2-4*i+1)*v[i-1]); v;

Formula

a(n) = Sum_{k=0..floor(n/2)} 3^k * |Stirling1(n,2*k)|.
a(n) = ( (sqrt(3))_n + (-sqrt(3))_n )/2, where (x)_n is the Pochhammer symbol.
a(0) = 1, a(1) = 0; a(n) = (2*n-3) * a(n-1) - (n^2-4*n+1) * a(n-2).
Showing 1-6 of 6 results.

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