cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A263687 b(n) in (sqrt(2))_n = b(n) + c(n)*sqrt(2), where (x)_n is the Pochhammer symbol, b(n) and c(n) are integers.

Original entry on oeis.org

1, 0, 2, 6, 26, 140, 896, 6636, 55804, 525168, 5468008, 62403880, 774616696, 10390122288, 149757486368, 2308301709840, 37887797229968, 659770432834688, 12148923787132832, 235858218326093664, 4814800618608693664, 103104123746671427520, 2310978427407268450048
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 23 2015

Keywords

Comments

The Pochhammer symbol (sqrt(2))_n = Gamma(n + sqrt(2))/Gamma(sqrt(2)) = sqrt(2)*(1 + sqrt(2))*(2 + sqrt(2))*...*(n - 1 + sqrt(2)).
(sqrt(2))_n = a(n) + A263688(n)*sqrt(2).

Examples

			For n = 4, (sqrt(2))_4 = sqrt(2)*(1 + sqrt(2))*(2 + sqrt(2))*(3 + sqrt(2)) = 26 + 18*sqrt(2), so a(4) = 26.
G.f. = 1 + 2*x^2 + 6*x^3 + 26*x^4 + 140*x^5 + 896*x^6 + 6636*x^7 + 55804*x^8 + ...

Links

Crossrefs

Cf. A263688.

Programs

  • Mathematica
    Expand@Table[(Pochhammer[Sqrt[2], n] + Pochhammer[-Sqrt[2], n])/2, {n, 0, 22}]
  • PARI
    {a(n) = if( n<0, 0, real(prod(k=0, n-1, quadgen(8) + k)))}; /* Michael Somos, Oct 23 2015 */

Formula

a(n) = ((sqrt(2))_n + (-sqrt(2))_n)/2.
E.g.f.: (1/(1-x)^sqrt(2)+(1-x)^sqrt(2))/2 = cosh(sqrt(2)*log(1-x)).
Recurrence: a(0) = 1, a(1) = 0, a(n+2) = (2*n+1)*a(n+1) + (2-n^2)*a(n).
a(n) ~ exp(-n)*n^(n+sqrt(2)-1/2)*sqrt(Pi/2)/Gamma(sqrt(2)).
0 = a(n)*(+7*a(n+1) - a(n+2) - 6*a(n+3) + a(n+4)) + a(n+1)*(+7*a(n+1) + 6*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) for all n>=0. - Michael Somos, Oct 23 2015
From Benedict W. J. Irwin, Oct 14 2016: (Start)
a(n) = (-1)^n*(binomial(-sqrt(2), n) + binomial(sqrt(2), n))*n!/2.
Conjecture: a(n) = (-1)^n * Sum_{k=0..n/2} Stirling1(n,2*k)*2^k.
(End)

A276474 a(n) = ((sqrt(2); sqrt(2))_n + (-sqrt(2); -sqrt(2))_n)/2, where (q; q)_n is the q-Pochhammer symbol.

Original entry on oeis.org

1, 1, -1, -5, 15, 87, -609, -8337, 125055, 2695455, -83559105, -4212669825, 265398198975, 22347926076735, -2838186611745345, -560679228377509185, 142973203236264842175, 47858338570309251530175, -24455611009428027531919425, -19225279650279123532147010625
Offset: 0

Views

Author

Vladimir Reshetnikov, Sep 12 2016

Keywords

Comments

The q-Pochhammer symbol (q; q)n = Product{k=1..n} (1 - q^k).

Links

Crossrefs

Cf. A276475, A263687, A263688.

Programs

  • Mathematica
    Round@Table[(QPochhammer[Sqrt[2], Sqrt[2], n] + QPochhammer[-Sqrt[2], -Sqrt[2], n])/2, {n, 0, 20}]

Formula

(sqrt(2); sqrt(2))_n = a(n) + A276475(n)*sqrt(2).
(-sqrt(2); -sqrt(2))_n = a(n) - A276475(n)*sqrt(2).

A263766 a(n) = Product_{k=1..n} (k^2 - 2).

Original entry on oeis.org

1, -1, -2, -14, -196, -4508, -153272, -7203784, -446634608, -35284134032, -3457845135136, -411483571081184, -58430667093528128, -9757921404619197376, -1893036752496124290944, -422147195806635716880512, -107225387734885472087650048
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 25 2015

Keywords

Examples

			For n = 3, a(3) = (1^2 - 2)*(2^2 - 2)*(3^2 - 2) = -14.
G.f. = 1 - x - 2*x^2 - 14*x^3 - 196*x^4 - 4508*x^5 - 153272*x^6 + ...

Links

Crossrefs

Cf. A101686, A263688, A263687.
Cf. A008865.

Programs

  • Haskell
    a263766 n = a263766_list !! n
a263766_list = scanl (*) 1 a008865_list
-- Reinhard Zumkeller, Oct 26 2015
  • Mathematica
    Table[Product[k^2 - 2, {k, 1, n}], {n, 0, 16}]
Expand@Table[-Pochhammer[Sqrt[2], n+1] Pochhammer[-Sqrt[2], n+1]/2, {n, 0, 16}]
Join[{1},FoldList[Times,Range[20]^2-2]] (* Harvey P. Dale, Aug 14 2022 *)
  • PARI
    a(n) = prod(k=1, n, k^2-2); \\ Michel Marcus, Oct 25 2015

Formula

a(n) = Gamma(1+sqrt(2)+n)*Gamma(1-sqrt(2)+n)*sin(Pi*sqrt(2))/(Pi*sqrt(2)).
a(n) = A263688(n+1)^2-A263687(n+1)^2/2.
a(n) ~ exp(-2*n)*n^(2*n+1)*sqrt(2)*sin(Pi*sqrt(2)).
G.f. for 1/a(n): hypergeom([1],[1-sqrt(2),1+sqrt(2)], x).
E.g.f. for 1/a(n): hypergeom([],[1-sqrt(2),1+sqrt(2)], x).
E.g.f. for a(n)/n!: hypergeom([1-sqrt(2),1+sqrt(2)], [1], x).
Recurrence: a(0) = 1, a(n) = (n^2-2)*a(n-1).
0 = a(n)*(-24*a(n+2) - 15*a(n+3) + a(n+4)) + a(n+1)*(-9*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) if n>=0. - Michael Somos, Oct 30 2015

A276475 a(n) = ((sqrt(2); sqrt(2))_n - (-sqrt(2); -sqrt(2))_n)/(2*sqrt(2)), where (q; q)_n is the q-Pochhammer symbol.

Original entry on oeis.org

0, -1, 1, 3, -9, -69, 483, 5355, -80325, -2081205, 64517355, 2738408715, -172519749045, -17158004483445, 2179066569397515, 365466952872801675, -93194072982564427125, -36694334101466364023925, 18750804725849312016225675
Offset: 0

Views

Author

Vladimir Reshetnikov, Sep 12 2016

Keywords

Comments

The q-Pochhammer symbol (q; q)n = Product{k=1..n} (1 - q^k).

Links

Crossrefs

Cf. A276474, A263687, A263688.

Programs

  • Mathematica
    Round@Table[(QPochhammer[Sqrt[2], Sqrt[2], n] - QPochhammer[-Sqrt[2], -Sqrt[2], n])/(2 Sqrt[2]), {n, 0, 20}]

Formula

(sqrt(2); sqrt(2))_n = A276474(n) + a(n)*sqrt(2).
(-sqrt(2); -sqrt(2))_n = A276474(n) - a(n)*sqrt(2).
Showing 1-4 of 4 results.

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