A357711 -id:A357711 - cp's OEIS frontend

A357712 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) * log(1-x) ).

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 12, 0, 1, 0, 4, 9, 26, 60, 0, 1, 0, 5, 12, 42, 140, 360, 0, 1, 0, 6, 15, 60, 240, 896, 2520, 0, 1, 0, 7, 18, 80, 360, 1614, 6636, 20160, 0, 1, 0, 8, 21, 102, 500, 2520, 12474, 55804, 181440, 0, 1, 0, 9, 24, 126, 660, 3620, 20160, 108900, 525168, 1814400, 0

Offset: 0

Seiichi Manyama, Oct 10 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, 12,  26,  42,  60,  80, ...
  0, 60, 140, 240, 360, 500, ...

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Columns k=0-4 give: A000007, (-1)^n * A105752(n), A263687, A357703, A357711.
Main diagonal gives A357683.
Cf. A357681.

T(n, k) = sum(j=0, n\2, k^j*abs(stirling(n, 2*j, 1)));

T(n, k) = round((prod(j=0, n-1, sqrt(k)+j)+prod(j=0, n-1, -sqrt(k)+j)))/2;

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

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).

cp's OEIS Frontend

A357712 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) * log(1-x) ).

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

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