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.

A047974 a(n) = a(n-1) + 2*(n-1)*a(n-2).

Original entry on oeis.org

1, 1, 3, 7, 25, 81, 331, 1303, 5937, 26785, 133651, 669351, 3609673, 19674097, 113525595, 664400311, 4070168161, 25330978113, 163716695587, 1075631907655, 7296866339961, 50322142646161, 356790528924523, 2570964805355607, 18983329135883665, 142389639792952801, 1091556096587136051
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Related to partially ordered sets. - Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003
The number of partial permutation matrices P in GL_n with P^2=0. Alternatively, the number of orbits of the Borel group of upper triangular matrices acting by conjugation on the set of matrices M in GL_n with M^2=0. - Brian Rothbach (rothbach(AT)math.berkeley.edu), Apr 16 2004
Number of ways to use the elements of {1..n} once each to form a collection of sequences, each having length 1 or 2. - Bob Proctor, Apr 18 2005
Hankel transform is A108400. - Paul Barry, Feb 11 2008
This is also the number of subsets of equivalent ways to arrange the elements of n pairs, when equivalence is defined under the joint operation of (optional) reversal of elements combined with permutation of the labels and the subset maps to itself. - Ross Drewe, Mar 16 2008
Equals inverse binomial transform of A000898. - Gary W. Adamson, Oct 06 2008
a(n) is also the moment of order n for the measure of density exp(-(x-1)^2/4)/(2*sqrt(Pi)) over the interval -oo..oo. - Groux Roland, Mar 26 2011
The n-th term gives the number of fixed-point-free involutions in S_n^B, the group of permutations on the set {-n,...,-1,1,2,...,n}. - Matt Watson, Jul 26 2012
From Peter Bala, Dec 03 2017: (Start)
a(n+k) == a(n) (mod k) for all n and k. Hence for each k, the sequence a(n) taken modulo k is a periodic sequence and the exact period divides k. Cf. A115329.
More generally, the same divisibility property holds for any sequence with an e.g.f. of the form F(x)*exp(x*G(x)), where F(x) and G(x) are power series with integer coefficients and G(0) = 1. See the Bala link for a proof. (End)

Links

Crossrefs

Row sums of A067147.
Column k=2 of A359762.
Cf. A000085, A000680, A000898, A001147, A001813, A100861, A108400, A115329, A132101.
Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), this sequence (q=1), A115329 (q=2), A293720 (q=4).

Programs

  • MATLAB
    N = 18; A = zeros(N,1); for n = 1:N; a = factorial(n); s = 0; k = 0; while k <= floor(n/2); b = factorial(n - 2*k); c = factorial(k); s = s + a/(b*c); k = k+1; end; A(n) = s; end; disp(A); % Ross Drewe, Mar 16 2008
  • Magma
    [n le 2 select 1 else Self(n-1) + 2*(n-2)*Self(n-2): n in [1..40]]; // G. C. Greubel, Jul 12 2024
  • Maple
    seq( add(n!/((n-2*k)!*k!), k=0..floor(n/2)), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 15 2001
with(combstruct):seq(count(([S,{S=Set(Union(Z,Prod(Z,Z)))},labeled],size=n)),n=0..30); # Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003
A047974 := n -> I^(-n)*orthopoly[H](n, I/2):
seq(A047974(n), n=0..26); # Peter Luschny, Nov 29 2017
  • Mathematica
    Range[0, 23]!*CoefficientList[ Series[ Exp[x*(1-x^2)/(1 - x)], {x, 0,23 }], x] - (* Zerinvary Lajos, Mar 23 2007 *)
Table[I^(-n)*HermiteH[n, I/2], {n, 0, 23}] - (* Alyssa Byrnes and C. Vignat, Jan 31 2013 *)
  • PARI
    my(x='x+O('x^66)); Vec(serlaplace(exp(x^2+x))) \\ Joerg Arndt, May 04 2013
  • SageMath
    [(-i)^n*hermite(n,i/2) for n in range(41)] # G. C. Greubel, Jul 12 2024

Formula

E.g.f.: exp(x^2+x). - Len Smiley, Dec 11 2001
Binomial transform of A001813 (with interpolated zeros). - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} C(k,n-k)*n!/k!. - Paul Barry, Mar 29 2007
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*(2k)!/k!; - Paul Barry, Feb 11 2008
G.f.: 1/(1-x-2*x^2/(1-x-4*x^2/(1-x-6*x^2/(1-x-8*x^2/(1-... (continued fraction). -Paul Barry, Apr 10 2009
E.g.f.: Q(0); Q(k) = 1+(x^2+x)/(2*k+1-(x^2+x)*(2*k+1)/((x^2+x)+(2*k+2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. Cf. A000085 and A115329. - Peter Bala, Dec 07 2011
a(n) ~ 2^(n/2 - 1/2)*exp(sqrt(n/2) - n/2 - 1/8)*n^(n/2). - Vaclav Kotesovec, Oct 08 2012
E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) = 1 + (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
a(n) = i^(-n)*H_{n}(i/2) with i the imaginary unit and H_{n} the Hermite polynomial of degree n. - Alyssa Byrnes and C. Vignat, Jan 31 2013
E.g.f.: -Q(0)/x where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
G.f.: 1/Q(0), where Q(k) = 1 + x*2*k - x/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013
E.g.f.: E(0)-1-x-x^2, where E(k) = 2 + 2*x*(1+x) - 8*k^2 + x^2*(1+x)^2*(2*k+3)*(2*k-1)/E(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 21 2013
E.g.f.: Product_{k>=1} 1/(1 + (-x)^k)^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019
a(n) = Sum_{k=0..floor(n/2)} 2^k*B(n, k), where B are the Bessel numbers A100861. - Peter Luschny, Jun 04 2021

A108400 a(n) = Product_{k = 0..n} (2^k * k!).

Original entry on oeis.org

1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000
Offset: 0

Views

Author

Philippe Deléham, Jul 02 2005

Keywords

Comments

Hankel transform (see A001906 for definition) of the sequences A000898, A001861, A035009(with first term omitted), A047974, A067147(unsigned version), A083886.
Hankel transform of the sequence with e.g.f. exp(x^2). Also (-1)^C(n+1,2)*a(n) is the Hankel transform of the sequence with e.g.f. exp(-x^2). - Paul Barry, Feb 12 2008
Let T(n,k) = (n+1)^k * (1+(-1)^(n-k))/2, then a(n) = det(T(i,j); 0<=i, j<=n). - Paul Barry, Feb 12 2008

Links

Crossrefs

Cf. A000178, A006125, A074962.
Cf. A000898, A001861, A001906, A035009, A047974, A067147, A083886.

Programs

  • Magma
    BarnesG:= func< n | (&*[Factorial(j): j in [0..n-2]]) >;
[2^Binomial(n+1,2)*BarnesG(n+2): n in [0..15]]; // G. C. Greubel, Jun 21 2022
  • Mathematica
    Table[Product[k!*2^k, {k,0,n}], {n,0,10}] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[2^Binomial[n+1,2]*BarnesG[n+2], {n,0,15}] (* G. C. Greubel, Jun 21 2022 *)
  • SageMath
    def barnes_g(n): return product(factorial(j) for j in (0..n-2))
[2^binomial(n+1,2)*barnes_g(n+2) for n in (0..15)] # G. C. Greubel, Jun 21 2022

Formula

a(n) = A006125(n+1)*A000178(n).
a(n) = Product_{i=1..n} Product_{j=0..i-1} {2*(i-j)}. - Paul Barry, Aug 02 2008
a(n) ~ 2^((n+1)^2/2) * n^(n^2/2+n+5/12) * Pi^((n+1)/2) / (A * exp(3*n^2/4+n-1/12)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

A001814 Coefficient of H_2 when expressing x^{2n} in terms of Hermite polynomials H_m.

Original entry on oeis.org

1, 12, 180, 3360, 75600, 1995840, 60540480, 2075673600, 79394515200, 3352212864000, 154872234316800, 7771770303897600, 420970891461120000, 24481076457277440000, 1521324036987955200000, 100610229646136770560000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) = A126804(n)/2. - Zerinvary Lajos, Sep 21 2007
a(n) is the number of ways to partition a set of 2n elements into parts of size 2 and then multiply by the number n of parts. - Alain Goupil, Jul 27 2025

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

a(n) = A048854(n, 1) = A067147(2n, 2).
Cf. A001879.
Cf. A005430.

Programs

  • Magma
    [Factorial(2*n)/(2*Factorial(n-1)): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011
  • Maple
    with(combinat):for n from 1 to 16 do printf(`%d, `,n!/2*sum(binomial(2*n, n), k=1..n)) od: # Zerinvary Lajos, Mar 13 2007
a:=n->sum((count(Permutation(n*2+2),size=n+1)),j=0..n)/2: seq(a(n), n=0..15); # Zerinvary Lajos, May 03 2007
seq(1/2*mul((n+k), k=1..n), n=0..16); # Zerinvary Lajos, Sep 21 2007
  • Mathematica
    Table[(2*n)!/(2*(n-1)!),{n,1,20}] (* Vincenzo Librandi, Nov 22 2011 *)
  • MuPAD
    combinat::catalan(n)*binomial(n+1,2)*n! $ n = 1..16; // Zerinvary Lajos, Feb 15 2007

Formula

E.g.f.: x/(1 - 4*x)^(3/2). - corrected by Alain Goupil, Jul 28 2025
a(n) = (2*n)!/(2*(n-1)!).
(n!/2)*binomial(2*n,n)*n or n!/2*A005430. - Zerinvary Lajos, Jun 06 2006
Sum_{n>=0} a(n)*x^(2n)/(2n)! = (x^2/2)*exp(x^2). - Alain Goupil, Jul 28 2025

Extensions

More terms and new description from Christian G. Bower, Dec 18 2001

A122832 Exponential Riordan array (e^(x(1+x)),x).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 25, 28, 18, 4, 1, 81, 125, 70, 30, 5, 1, 331, 486, 375, 140, 45, 6, 1, 1303, 2317, 1701, 875, 245, 63, 7, 1, 5937, 10424, 9268, 4536, 1750, 392, 84, 8, 1, 26785, 53433, 46908, 27804, 10206, 3150, 588, 108, 9, 1
Offset: 0

Views

Author

Paul Barry, Sep 12 2006

Keywords

Comments

Row sums are A000898. Inverse is A122833. Product of A007318 and A067147.

Examples

			Triangle begins:
   1;
   1,   1;
   3,   2,  1;
   7,   9,  3,  1;
  25,  28, 18,  4, 1;
  81, 125, 70, 30, 5, 1;
  ...
From _Peter Bala_, May 14 2012: (Start)
T(3,1) = 9. The 9 ways to select a subset of {1,2,3} of size 1 and arrange the remaining elements into a set of lists (denoted by square brackets) of length 1 or 2 are:
{1}[2,3], {1}[3,2], {1}[2][3],
{2}[1,3], {2}[3,1], {2}[1][3],
{3}[1,2], {3}[2,1], {3}[1][2]. (End)

Links

Crossrefs

A000898 (row sums), A047974 (column 0), A291632 (column 1), A122833 (inverse array).
Cf. A059344, A111062.

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
RiordanArray[E^(#(1+#))&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)
  • PARI
    T(n,k) = (n!/k!)*sum(i=0, n-k, binomial(i,n-k-i)/i!); \\ Michel Marcus, Aug 28 2017

Formula

Number triangle T(n,k) = (n!/k!)*Sum_{i = 0..n-k} C(i,n-k-i)/i!.
From Peter Bala, May 14 2012: (Start)
Array is exp(S + S^2) where S is A132440 the infinitesimal generator for Pascal's triangle.
T(n,k) = binomial(n,k)*A047974(n-k).
So T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then arrange the remaining n-k elements into a set of lists of length 1 or 2. (End)
From Peter Bala, Oct 24 2023: (Start)
n-th row polynomial: R(n,x) = exp(D + D^2) (x^n) = exp(D^2) (1 + x)^n, where D denotes the derivative operator d/dx. Cf. A111062.
The sequence of polynomials defined by R(n,x-1) = exp(D^2) (x^n) begins [1, 1, 2 + x^2, 6*x + x^3, 12 + 12*x^2 + x^4, ...] and is related to the Hermite polynomials. See A059344. (End)

Extensions

More terms from Michel Marcus, Aug 28 2017

A112227 A scaled Hermite triangle.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, 0, -6, 0, 1, 12, 0, -12, 0, 1, 0, 60, 0, -20, 0, 1, -120, 0, 180, 0, -30, 0, 1, 0, -840, 0, 420, 0, -42, 0, 1, 1680, 0, -3360, 0, 840, 0, -56, 0, 1, 0, 15120, 0, -10080, 0, 1512, 0, -72, 0, 1, -30240, 0, 75600, 0, -25200, 0, 2520, 0, -90, 0, 1, 0, -332640, 0, 277200, 0, -55440, 0, 3960, 0, -110, 0, 1, 665280, 0
Offset: 0

Views

Author

Paul Barry, Aug 28 2005

Keywords

Comments

Inverse of number triangle A067147. Diagonal sums are A002119.

Examples

			Triangle begins
1;
0,1;
-2,0,1;
0,-6,0,1;
12,0,-12,0,1;
0,60,0,-20,0,1;

Links

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
rows = 12;
R = RiordanArray[E^(-#^2)&, #&, rows, True];
R // Flatten

Formula

Number triangle T(n, k)=A060821(n, k)/2^k; T(n, k)=n!/(k!*2^((n-k)/2)((n-k)/2)!)*cos(pi*(n-k)/2)*2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1) T(n, k)=A001498((n+k)/2, (n-k)/2)*cos(pi(n-k)/2)*2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1);
Exponential Riordan array (e^(-x^2),x). - Paul Barry, Sep 12 2006

A340262 T(n, k) = multinomial(n + k/2; n, k/2) if k is even else 0. Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 0, 3, 1, 0, 4, 0, 1, 0, 5, 0, 15, 1, 0, 6, 0, 21, 0, 1, 0, 7, 0, 28, 0, 84, 1, 0, 8, 0, 36, 0, 120, 0, 1, 0, 9, 0, 45, 0, 165, 0, 495, 1, 0, 10, 0, 55, 0, 220, 0, 715, 0, 1, 0, 11, 0, 66, 0, 286, 0, 1001, 0, 3003, 1, 0, 12, 0, 78, 0, 364, 0, 1365, 0, 4368, 0
Offset: 0

Views

Author

Peter Luschny, Jan 05 2021

Keywords

Examples

			Triangle starts:
                             [0] 1;
                           [1] 1, 0;
                          [2] 1, 0, 3;
                        [3] 1, 0, 4, 0;
                      [4] 1, 0, 5, 0, 15;
                     [5] 1, 0, 6, 0, 21, 0;
                   [6] 1, 0, 7, 0, 28, 0, 84;
                 [7] 1, 0, 8, 0, 36, 0, 120, 0;
              [8] 1, 0, 9, 0, 45, 0, 165, 0, 495;
            [9] 1, 0, 10, 0, 55, 0, 220, 0, 715, 0;

Crossrefs

Cf. A130746, A067147.

Programs

  • Maple
    T := proc(n, k) `if`(k::even, combinat:-multinomial(n + k/2, n, k/2), 0) end:
seq(seq(T(n,k), k=0..n), n=0..11);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
T[n_, k_] := If[EvenQ[k], multinomial[n + k/2, {n, k/2}], 0];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 18 2024 *)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.