A357725 -id:A357725 - cp's OEIS frontend

A357728 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * (exp(x) - 1) ).

1, 1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, -2, -3, 0, 1, 0, -3, -6, -6, 0, 1, 0, -4, -9, -10, -5, 0, 1, 0, -5, -12, -12, 10, 33, 0, 1, 0, -6, -15, -12, 45, 190, 266, 0, 1, 0, -7, -18, -10, 100, 465, 1106, 1309, 0, 1, 0, -8, -21, -6, 175, 852, 2394, 4438, 4905, 0, 1, 0, -9, -24, 0, 270, 1345, 4004, 7827, 9978, 11516, 0

Offset: 0

Seiichi Manyama, Oct 11 2022

			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, -6, -10, -12, -12, -10, ...
  0, -5,  10,  45, 100, 175, ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Bell Polynomial.

Columns k=0-4 give: A000007, A121867, A357725, A357726, A357727.
Main diagonal gives A357729.
Cf. A357681, A357720.

T(n, k) = sum(j=0, n\2, (-k)^j*stirling(n, 2*j, 2));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
T(n, k) = round((Bell_poly(n, sqrt(k)*I)+Bell_poly(n, -sqrt(k)*I)))/2;

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

T(n,k) = ( Bell_n(sqrt(k) * i) + Bell_n(-sqrt(k) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

A357693 Expansion of e.g.f. cos( sqrt(2) * log(1+x) ).

Original entry on oeis.org

1, 0, -2, 6, -18, 60, -216, 756, -1620, -14256, 349272, -5452920, 78885576, -1143659088, 17074183104, -265437239760, 4316991698448, -73572489226368, 1314108286270560, -24584195654596512, 481215937895868384, -9843358555320333120, 210128893733994567552

Offset: 0

Seiichi Manyama, Oct 10 2022

sign

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Column k=2 of A357720.
Cf. A003703, A357718, A357719.
Cf. A357725.

With[{nn=30},CoefficientList[Series[Cos[Sqrt[2]Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 04 2024 *)

my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*log(1+x)))))

a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 1));

a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(2)*I+k)+prod(k=0, n-1, -sqrt(2)*I+k)))/2;

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+6)*v[i-1]); v;

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

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

Seiichi Manyama, Oct 11 2022

sign

Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A264037, A357725.

my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sin(sqrt(2)*(exp(x)-1))/sqrt(2)))))

a(n) = sum(k=0, (n-1)\2, (-2)^k*stirling(n, 2*k+1, 2));

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));

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.

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A357728 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * (exp(x) - 1) ).

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A357693 Expansion of e.g.f. cos( sqrt(2) * log(1+x) ).

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

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