A357726 -id:A357726 - 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.

A357718 Expansion of e.g.f. cos( sqrt(3) * log(1+x) ).

Original entry on oeis.org

1, 0, -3, 9, -24, 60, -84, -756, 13104, -157248, 1795248, -20900880, 254007936, -3250473408, 43922668608, -626830626240, 9437477107968, -149644407564288, 2493958878657792, -43592393744250624, 797394015216175104, -15230735270523601920

Offset: 0

Seiichi Manyama, Oct 10 2022

sign

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Column k=3 of A357720.

Cf. A357703, A357726.

my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(3)*log(1+x)))))

a(n) = sum(k=0, n\2, (-3)^k*stirling(n, 2*k, 1));

a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(3)*I+k)+prod(k=0, n-1, -sqrt(3)*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+7)*v[i-1]); v;

a(n) = Sum_{k=0..floor(n/2)} (-3)^k * Stirling1(n,2*k).
a(n) = (-1)^n * ( (sqrt(3) * i)_n + (-sqrt(3) * 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+7) * a(n-2).

A357737 Expansion of e.g.f. sin( sqrt(3) * (exp(x) - 1) )/sqrt(3).

Original entry on oeis.org

0, 1, 1, -2, -17, -65, -134, 331, 5797, 41092, 199621, 500731, -2996432, -58995155, -573624323, -4065029714, -19194210269, 7657775035, 1581081323122, 24363365708815, 260409006907921, 2127851409822892, 11143555796154673, -27234657667343081

Offset: 0

Seiichi Manyama, Oct 11 2022

sign

Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A357572, A357726.

my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sin(sqrt(3)*(exp(x)-1))/sqrt(3)))))

a(n) = sum(k=0, (n-1)\2, (-3)^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(3)*I)-Bell_poly(n, -sqrt(3)*I))/(2*sqrt(3)*I));

a(n) = Sum_{k=0..floor((n-1)/2)} (-3)^(k) * Stirling2(n,2*k+1).
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357726(k).
a(n) = ( Bell_n(sqrt(3) * i) - Bell_n(-sqrt(3) * i) )/(2 * sqrt(3) * i), where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

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) ).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A357718 Expansion of e.g.f. cos( sqrt(3) * log(1+x) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

PARI

PARI

Formula

A357737 Expansion of e.g.f. sin( sqrt(3) * (exp(x) - 1) )/sqrt(3).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula