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 1-6 of 6 results.

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).

A263688 c(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

0, 1, 1, 4, 18, 98, 630, 4676, 39368, 370748, 3861900, 44087008, 547360968, 7342948312, 105848450344, 1631635791184, 26782838577600, 466413214471568, 8588795078851344, 166747235206457024, 3404055687248777120, 72895914363584236064, 1633918325381940384864
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 = A263687(n) + a(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) = 18.
G.f. = x + x^2 + 4*x^3 + 18*x^4 + 98*x^5 + 630*x^6 + 4676*x^7 + 39368*x^8 + ...

Links

Crossrefs

Cf. A263687.

Programs

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

Formula

a(n) = ((sqrt(2))_n - (-sqrt(2))_n)/(2*sqrt(2)).
E.g.f.: (1/(1-x)^sqrt(2)-(1-x)^sqrt(2))/(2*sqrt(2)) = -sinh(sqrt(2)*log(1-x))/sqrt(2).
D-finite with recurrence: a(0) = 0, a(1) = 1, 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

A357683 a(n) = Sum_{k=0..floor(n/2)} n^k * |Stirling1(n,2*k)|.

Original entry on oeis.org

1, 0, 2, 9, 60, 500, 4920, 55566, 706720, 9979200, 154706760, 2609691700, 47547916416, 929943488448, 19421810408000, 431196538865400, 10137091700736000, 251485260368396288, 6563768030597826720, 179746132716715050000, 5152012082327932518400
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2022

Keywords

Links

Crossrefs

Cf. A105752, A263687.

Programs

  • PARI
    a(n) = sum(k=0, n\2, n^k*abs(stirling(n, 2*k, 1)));
  • PARI
    a(n) = round(n!*polcoef(cosh(sqrt(n)*log(1-x+x*O(x^n))), n));
  • PARI
    a(n) = round((prod(k=0, n-1, sqrt(n)+k)+prod(k=0, n-1, -sqrt(n)+k)))/2;

Formula

a(n) = n! * [x^n] cosh( sqrt(n) * log(1-x) ).
a(n) = ( (sqrt(n))_n + (-sqrt(n))_n )/2, where (x)_n is the Pochhammer symbol.
a(n) ~ n^(n + sqrt(n)/2 - 1/4) / (2*exp(n - sqrt(n) - 1/2)) * (1 - 3/(4*sqrt(n))). - Vaclav Kotesovec, Oct 10 2022

A357712 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * log(1-x) ).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 12, 0, 1, 0, 4, 9, 26, 60, 0, 1, 0, 5, 12, 42, 140, 360, 0, 1, 0, 6, 15, 60, 240, 896, 2520, 0, 1, 0, 7, 18, 80, 360, 1614, 6636, 20160, 0, 1, 0, 8, 21, 102, 500, 2520, 12474, 55804, 181440, 0, 1, 0, 9, 24, 126, 660, 3620, 20160, 108900, 525168, 1814400, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Examples

			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, 12,  26,  42,  60,  80, ...
  0, 60, 140, 240, 360, 500, ...

Links

Crossrefs

Columns k=0-4 give: A000007, (-1)^n * A105752(n), A263687, A357703, A357711.
Main diagonal gives A357683.
Cf. A357681.

Programs

  • PARI
    T(n, k) = sum(j=0, n\2, k^j*abs(stirling(n, 2*j, 1)));
  • PARI
    T(n, k) = round((prod(j=0, n-1, sqrt(k)+j)+prod(j=0, n-1, -sqrt(k)+j)))/2;

Formula

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

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 1-6 of 6 results.

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