A357719 -id:A357719 - cp's OEIS frontend

A357720 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) * log(1+x) ).

1, 1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, -2, 3, 0, 1, 0, -3, 6, -10, 0, 1, 0, -4, 9, -18, 40, 0, 1, 0, -5, 12, -24, 60, -190, 0, 1, 0, -6, 15, -28, 60, -216, 1050, 0, 1, 0, -7, 18, -30, 40, -84, 756, -6620, 0, 1, 0, -8, 21, -30, 0, 200, -756, -1620, 46800, 0, 1, 0, -9, 24, -28, -60, 630, -3360, 13104, -14256, -365300, 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, -10, -18, -24, -28, -30, ...
  0,  40,  60,  60,  40,   0, ...

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Columns k=0-4 give: A000007, (-1)^n * A003703, A357693, A357718, A357719.
Main diagonal gives A357721.
Cf. A357712, A357728.

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

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

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

A357727 Expansion of e.g.f. cos( 2 * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, -4, -12, -12, 100, 852, 4004, 9940, -36828, -726316, -6174300, -35968812, -109708508, 702818004, 16677814436, 188794428628, 1542659688996, 8359981681364, -3068614764636, -868989327994668, -15076627082974940, -179727483880747308

Offset: 0

Seiichi Manyama, Oct 10 2022

sign

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Bell Polynomial.

Column k=4 of A357728.

Cf. A065143, A357719, A357738.

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

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

a(n) = Sum_{k=0..floor(n/2)} (-4)^k * Stirling2(n,2*k).
a(n) = 1; a(n) = -4 * Sum_{k=0..n-1} binomial(n-1, k) * A357738(k).
a(n) = ( Bell_n(2 * i) + Bell_n(-2 * 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).

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A357720 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) * log(1+x) ).

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A357727 Expansion of e.g.f. cos( 2 * (exp(x) - 1) ).

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

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