A001464 -id:A001464 - cp's OEIS frontend

A362718 Expansion of e.g.f. cos(x)exp(x^2/2) = Sum_{n>=0} a(n)x^(2n)/(2n)!.

1, 0, -2, -16, -132, -1216, -12440, -138048, -1601264, -18108928, -161934624, 404007680, 92590134208, 4221314202624, 159324751301248, 5730872535686144, 205239818509082880, 7450322829180649472, 276342876017093172736, 10509280308463090102272

Offset: 0

Michael Somos, Apr 30 2023

sign

Cf. A001464.

a[ n_] := If[ n<0, 0, (2*n)! * SeriesCoefficient[ Cos[x] * Exp[x^2/2], {x, 0, 2*n}]];

{a(n) = my(A); if( n<0, 0, A = x*O(x^(2*n)); (2*n)! * polcoef( cos(x + A)*exp(x^2/2 + A), 2*n))};

def egfExpand(f, step, size) -> list[int]:
    x = LazyPowerSeriesRing(QQ, "x").gen()
    return [f(x)[step*n] * factorial(step*n) for n in range(size+1)]
def egf(x): return cos(x)*exp(x^2/2)
print(egfExpand(egf, 2, 19))  # Peter Luschny, May 02 2023

a(n) = (-1)^n * A001464(2*n).

0 = a(n)*(360*a(n+2) -600*a(n+3) +230*a(n+4) -28*a(n+5) +a(n+6)) +a(n+1)*(216*a(n+2) -296*a(n+3) +84*a(n+4) -6*a(n+5)) +a(n+2)*(66*a(n+2) -56*a(n+3) +15*a(n+4)) -10*a(n+3)^2 for all n >= 0.

A295343 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j/j!).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, 0, -1, 0, 1, -1, 0, 2, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -1, 0, 1, 2, -6, 1, 0, 1, -1, 0, 1, 1, -6, 16, -1, 0, 1, -1, 0, 1, 1, -1, -14, 20, 1, 0, 1, -1, 0, 1, 1, -2, -14, 20, -132, -1, 0, 1, -1, 0, 1, 1, -2, -8, -15, 204, -28, 1, 0, 1, -1, 0, 1, 1, -2, -9, -15, 99, 28, 1216, -1, 0

Offset: 0

Ilya Gutkovskiy, Nov 20 2017

			Square array begins:
1,  1,  1,  1,  1,  1,  ...
0, -1  -1, -1, -1, -1,  ...
0,  1,  0,  0,  0,  0,  ...
0, -1,  2,  1,  1,  1,  ...
0,  1, -2,  2,  1,  1,  ...
0, -1, -6, -6, -1, -2,  ...

Columns k=0..3 give A000007, A033999, A001464, A014775.
Main diagonal gives A000587.
Cf. A229223.

Table[Function[k, n! SeriesCoefficient[Exp[-Sum[x^i/i!, {i, 1, k}]], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[1 - Exp[x] Gamma[k + 1, x]/Gamma[k + 1]], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(-Sum_{j=1..k} x^j/j!).

cp's OEIS Frontend

A362718 Expansion of e.g.f. cos(x)exp(x^2/2) = Sum_{n>=0} a(n)x^(2n)/(2n)!.

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A295343 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j/j!).

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cp's OEIS Frontend

A362718 Expansion of e.g.f. cos(x)*exp(x^2/2) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

SageMath

Formula

A295343 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j/j!).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A362718 Expansion of e.g.f. cos(x)exp(x^2/2) = Sum_{n>=0} a(n)x^(2n)/(2n)!.