A047991 -id:A047991 - cp's OEIS frontend

A047990 a(n+1) = a(n) + (n^2 + 1)*a(n-1).

1, 2, 4, 14, 54, 292, 1696, 12500, 97300, 909800, 8888400, 100778200, 1185163000, 15798002000, 217275712000, 3329482106000, 52433793018000, 908110694260000, 16113910669480000, 311249886303980000, 6144485548655740000

Offset: 0

N. J. A. Sloane

Studied by M. Bhargava (bhargava(AT)math.Princeton.EDU) and H. S. Wilf.

T. D. Noe, Table of n, a(n) for n=0..100

Cf. A047991.

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==a[n-1]+((n-1)^2+1)a[n-2]},a,{n,20}] (* Harvey P. Dale, May 07 2013 *)

a(n) ~ c * n!, where c = sqrt(Pi) * (1/(Gamma(1/2 + i/2) * Gamma(1 - i/2)) + 1/(Gamma(1/2 - i/2) * Gamma(1 + i/2)))/2 + hypergeom([1 - i, 1 - i], [2], -1) / 2^i = 2.6068398732089369139523868268495087173767147472699250704975984714... and i is the imaginary unit. - Vaclav Kotesovec, Sep 26 2021

A348023 a(n) = [x^n] Product_{k=0..2n-1} (x + (-1)^k k).

Original entry on oeis.org

1, -1, -5, 51, 1009, -17765, -636385, 15875083, 828730833, -26734633497, -1837389309405, 72689703927555, 6202163408816881, -290609121018808813, -29631450442239587305, 1604436906627413556075, 190333687041180914380065, -11692931012450127257939505, -1582217617564533519968758645

Offset: 0

Seiichi Manyama, Sep 25 2021

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Seiichi Manyama, Table of n, a(n) for n = 0..351

Cf. A047991, A187646, A348024.

a(n) = polcoef(prod(k=0, 2*n-1, x+(-1)^k*k), n);

a(n) = A047991(2*n-1,n) for n>0.

A348024 a(n) = [x^n] Product_{k=1..2n} (x + (-1)^k k).

Original entry on oeis.org

1, 1, -13, -87, 2609, 34965, -1638841, -33411007, 2128380881, 58501278297, -4709853961605, -163166745402375, 15875893807564465, 664163782026582877, -75766955207796981345, -3716140360596415682175, 486265092408065178623265, 27363821966293458296896305, -4039473192629570940395992765

Offset: 0

Seiichi Manyama, Sep 25 2021

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Seiichi Manyama, Table of n, a(n) for n = 0..350

Cf. A047991, A129505, A348023.

a(n) = polcoef(prod(k=1, 2*n, x+(-1)^k*k), n);

a(n) = A047991(2*n,n).

cp's OEIS Frontend

A047990 a(n+1) = a(n) + (n^2 + 1)*a(n-1).

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Formula

A348023 a(n) = [x^n] Product_{k=0..2n-1} (x + (-1)^k k).

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Author

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PARI

Formula

A348024 a(n) = [x^n] Product_{k=1..2n} (x + (-1)^k k).

Views

Author

Keywords

Links

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Programs

PARI

Formula

cp's OEIS Frontend

A047990 a(n+1) = a(n) + (n^2 + 1)*a(n-1).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A348023 a(n) = [x^n] Product_{k=0..2*n-1} (x + (-1)^k * k).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A348024 a(n) = [x^n] Product_{k=1..2*n} (x + (-1)^k * k).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A348023 a(n) = [x^n] Product_{k=0..2n-1} (x + (-1)^k k).

A348024 a(n) = [x^n] Product_{k=1..2n} (x + (-1)^k k).