A357727 -id:A357727 - 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.

A357719 Expansion of e.g.f. cos( 2 * log(1+x) ).

Original entry on oeis.org

1, 0, -4, 12, -28, 40, 200, -3360, 35680, -357120, 3644800, -38896000, 437756800, -5206406400, 65372153600, -864339840000, 11991424640000, -173800340480000, 2617640829440000, -40693929269760000, 647089190924800000, -10383194262604800000

Offset: 0

Seiichi Manyama, Oct 10 2022

sign

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Column k=4 of A357720.

Cf. A357711, A357727.

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

a(n) = sum(k=0, n\2, (-4)^k*stirling(n, 2*k, 1));

a(n) = (-1)^n*(prod(k=0, n-1, 2*I+k)+prod(k=0, n-1, -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+8)*v[i-1]); v;

a(n) = Sum_{k=0..floor(n/2)} (-4)^k * Stirling1(n,2*k).
a(n) = (-1)^n * ( (2 * i)_n + (-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+8) * a(n-2).

A357738 Expansion of e.g.f. sin( 2 * (exp(x) - 1) )/2.

Original entry on oeis.org

0, 1, 1, -3, -23, -83, -119, 973, 11145, 69805, 278281, 33165, -12794231, -157150355, -1271714807, -7108146611, -11364216951, 380051588653, 6923479542025, 78935931180813, 669998027706505, 3602978599128301, -8825050911646199, -598024924863875123

Offset: 0

Seiichi Manyama, Oct 11 2022

sign

Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A357598, A357727.

With[{nn=30},CoefficientList[Series[Sin[2(Exp[x]-1)]/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 19 2023 *)

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

a(n) = sum(k=0, (n-1)\2, (-4)^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, 2*I)-Bell_poly(n, -2*I)))/(4*I);

a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^(k) * Stirling2(n,2*k+1).
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357727(k).
a(n) = ( Bell_n(2 * i) - Bell_n(-2 * i) )/(4 * 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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A357719 Expansion of e.g.f. cos( 2 * log(1+x) ).

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A357738 Expansion of e.g.f. sin( 2 * (exp(x) - 1) )/2.

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