A376826 -id:A376826 - cp's OEIS frontend

A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.

1, 2, 6, 20, 76, 312, 1384, 6512, 32400, 168992, 921184, 5222208, 30710464, 186753920, 1171979904, 7573069568, 50305536256, 342949298688, 2396286830080, 17138748412928, 125336396368896, 936222729254912, 7136574106003456, 55466948299223040, 439216305474605056, 3540846129311916032

Offset: 0

N. J. A. Sloane

Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).
Also the value of the n-th derivative of exp(x^2) evaluated at 1. - N. Calkin, Apr 22 2010
For n >= 1, a(n) is also the sum of the degrees of the irreducible representations of the group of n X n signed permutation matrices (described in sequence A066051). The similar sum for the "ordinary" symmetric group S_n is in sequence A000085. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 12 2002
It appears that this is also the number of permutations of 1, 2, ..., n+1 such that each term (after the first) is within 2 of some preceding term. Verified for n+1 <= 6. E.g., a(4) = 20 because of the 24 permutations of 1, 2, 3, 4, the only ones not permitted are 1, 4, 2, 3; 1, 4, 3, 2; 4, 1, 2, 3; and 4, 1, 3, 2. - Gerry Myerson, Aug 06 2003
Hankel transform is A108400. - Paul Barry, Feb 11 2008
From Emeric Deutsch, Jun 19 2010: (Start)
Number of symmetric involutions of [2n]. Example: a(2)=6 because we have 1234, 2143, 1324, 3412, 4231, and 4321. See the Egge reference, pp. 419-420.
Number of symmetric involutions of [2n+1]. Example: a(2)=6 because we have 12345, 14325, 21354, 45312, 52341, and 54321. See the Egge reference, pp. 419-420.
(End)
Binomial convolution of sequence A000085: a(n) = Sum_{k=0..n} binomial(n,k)*A000085(k)*A000085(n-k). - Emanuele Munarini, Mar 02 2016
The sequence can be obtained from the infinite product of 2 X 2 matrices [(1,N); (1,1)] by extracting the upper left terms, where N = (1, 3, 5, ...), the odd integers. - Gary W. Adamson, Jul 28 2016
Apparently a(n) is the number of standard domino tableaux of size 2n, where a domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. - Gus Wiseman, Feb 25 2018

			G.f. = 1 + 2*x + 6*x^2 + 20*x^3 + 76*x^4 + 312*x^5 + 1384*x^6 + 6512*x^7 + ...
The a(3) = 20 domino tableaux:
1 1 2 2 3 3
.
1 2 2 3 3
1
.
1 2 3 3   1 1 3 3   1 1 2 2
1 2       2 2       3 3
.
1 1 3 3   1 1 2 2
2         3
2         3
.
1 2 3   1 2 2   1 1 3
1 2 3   1 3 3   2 2 3
.
1 3 3   1 2 2
1       1
2       3
2       3
.
1 2   1 1   1 1
1 2   2 3   2 2
3 3   2 3   3 3
.
1 3   1 2   1 1
1 3   1 2   2 2
2     3     3
2     3     3
.
1 1
2
2
3
3
.
1
1
2
2
3
3 - _Gus Wiseman_, Feb 25 2018

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.1.4 Exer. 31.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
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).

T. D. Noe, Table of n, a(n) for n=0..200
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 10.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 10.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
R. A. Brualdi, Shi-Mie Ma, Enumeration of involutions by descents and symmetric matrices, Eur. J. Combin. 43 (2015) 220-228
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C.-O. Chow, Counting involutory, unimodal, and alternating signed permutations, Discr. Math. 306 (2006), 2222-2228.
R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405-434.
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv:1302.6150 [math.RT], 2013-2014.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Yen-chi R. Lin, Asymptotic Formula for Symmetric Involutions, arXiv:1310.0988 [math.CO], 2013.
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
D. P. Roberts and A. Venkatesh, Hurwitz monodromy and full number fields, 2014.
Index entries for sequences related to Hermite polynomials

Column k=2 of A294042 and A376826.
Row sums of A122832 and A185296.
Cf. A000085, A000712, A004003, A066051, A099390, A100861, A135401, A138178, A153452, A297388, A299699, A299926, A300056, A300060.

a000898 n = a000898_list !! n
a000898_list = 1 : 2 : (map (* 2) $
   zipWith (+) (tail a000898_list) (zipWith (*) [1..] a000898_list))
-- Reinhard Zumkeller, Oct 10 2011

# For Maple program see A000903.
seq(simplify((-I)^n*HermiteH(n, I)), n=0..25); # Peter Luschny, Oct 23 2015

a[n_] := Sum[ 2^k*StirlingS1[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 17 2011, after Vladeta Jovovic *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2(a[n-1]+(n-1)a[n-2])},a,{n,30}] (* Harvey P. Dale, Aug 04 2012 *)
Table[Abs[HermiteH[n, I]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
a[ n_] := Sum[ 2^(n - 2 k) n! / (k! (n - 2 k)!), {k, 0, n/2}]; (* Michael Somos, Oct 23 2015 *)

makelist((%i)^n*hermite(n,-%i),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp(2*x + x^2 + x * O(x^n)), n))}; /* Michael Somos, Feb 08 2004 */

{a(n) = if( n<2, max(0, n+1), 2*a(n-1) + (2*n - 2) * a(n-2))}; /* Michael Somos, Feb 08 2004 */

my(x='x+O('x^66)); Vec(serlaplace(exp(2*x+x^2))) \\ Joerg Arndt, Oct 04 2013

{a(n) = sum(k=0, n\2, 2^(n - 2*k) * n! / (k! * (n - 2*k)!))}; /* Michael Somos, Oct 23 2015 */

a(n) = Sum_{m=0..n} |A060821(n,m)| = H(n,-i)*i^n, with the Hermite polynomials H(n,x); i.e., these are row sums of the unsigned triangle A060821.
E.g.f.: exp(x*(x + 2)).
a(n) = 2 * A000902(n) for n >= 1.
a(n) = Sum_{k=0..n} binomial(n,2k)*binomial(2k,k)*k!*2^(n-2k). - N. Calkin, Apr 22 2010
Binomial transform of A047974. - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k). - Vladeta Jovovic, Oct 01 2003
From Paul Barry, Aug 29 2005: (Start)
a(n) = Sum_{k=0..floor(n/2)} A001498(n-k, k) * 2^(n-k).
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2) * 2^((n+k)/2) * (1+(-1)^(n-k))/2. (End)
For asymptotics, see the Robinson paper. [This is disputed by Yen-chi R. Lin. See below, Sep 30 2013.]
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * C(n,2*k) * (2*k)!/k!. - Paul Barry, Feb 11 2008
G.f.: 1/(1 - 2*x - 2*x^2/(1 - 2*x - 4*x^2/(1 - 2*x - 6*x^2/(1 - 2*x - 8*x^2/(1 - ... (continued fraction). - Paul Barry, Feb 25 2010
E.g.f.: exp(x^2 + 2*x) = Q(0); Q(k) = 1 + (x^2 + 2*x)/(2*k + 1 - (x^2 + 2*x)*(2*k + 1)/((x^2 + 2*x) + (2*k + 2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: 1/Q(0), where Q(k) = 1 + 2*x*k - x - x/(1 - 2*x*(k + 1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
a(n) = (2*n/e)^(n/2) * exp(sqrt(2*n)) / sqrt(2*e) * (1 + sqrt(2/n)/3 + O(n^(-1))). - Yen-chi R. Lin, Sep 30 2013
0 = a(n)*(2*a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(-2*a(n+1) + a(n+2)) for all n >= 0. - Michael Somos, Oct 23 2015
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k)*B(n, k), where B are the Bessel numbers A100861. - Peter Luschny, Jun 04 2021

More terms from Larry Reeves (larryr(AT)acm.org), Feb 21 2001
Initial condition a(0)=1 added to definition by Jon E. Schoenfield, Oct 01 2013
More terms from Joerg Arndt, Oct 04 2013

A193778 Number of signed permutations of length 2n invariant under D and D'bar.

Original entry on oeis.org

1, 2, 8, 32, 160, 832, 4864, 29696, 195584, 1341440, 9723904, 73105408, 574062592, 4657184768, 39165624320, 339133595648, 3028204650496, 27760959422464, 261439835078656, 2521668748574720, 24912764963127296, 251559029812232192, 2595790316527157248

Offset: 0

Andy Hardt, Aug 04 2011

nonn

See section 6 in the Hardt reference.

			For n = 1, the permutations of length 2 are (-1, 2) and (1, -2).

Harvey P. Dale, Table of n, a(n) for n = 0..661
Andy Hardt and Justin M. Troyka, Restricted Symmetric Signed Permutations, Department of Mathematics, Carleton College, 2013.

Column k=4 of A376826.

Cf. A193777, A000085.

exp(2*x*(x + 1)): series(%, x, 23):
seq(n!*coeff(%, x, n), n = 0..22); # Peter Luschny, May 30 2021

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2a[n-1]+4(n-1)a[n-2]},a,{n,30}] (* Harvey P. Dale, Mar 14 2021 *)
a[n_] := 2^n HypergeometricPFQ[{-n/2, (1-n)/2}, {}, 2];
Table[a[n], {n, 0, 22}] (* Peter Luschny, May 30 2021 *)

seq(n)={my(v=vector(n+1)); v[1]=1; v[2]=2; for(n=2, n, v[n+1] = 2*v[n] + 4*(n-1)*v[n-1]); v} \\ Andrew Howroyd, Dec 08 2018

a(n) = 2*a(n-1) + 4*(n - 1)*a(n-2).
From Peter Luschny, May 30 2021: (Start)
a(n) = 2^n * hypergeom([-n/2, (1 - n)/2], [], 2).
a(n) = 2^n * A000085(n).
a(n) = n! * [x^n] exp(2*x*(x + 1)). (End)

Terms a(6) and beyond from Andrew Howroyd, Dec 08 2018

A202830 E.g.f.: exp(2x + 3x^2/2).

Original entry on oeis.org

1, 2, 7, 26, 115, 542, 2809, 15374, 89737, 548450, 3519799, 23493098, 163139563, 1172030654, 8706504265, 66638295998, 525069283921, 4248776775746, 35276087031463, 299986119953210, 2610709200699811, 23220585598592222, 210915850841272537, 1954390351189631726

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

From Emanuele Munarini, May 16 2014: (Start)
a(n) is the number of partitions of an n-set where each block consists of one or two elements, and each block is endowed with a permutation or a linear order.
For instance, for n = 2, we have the following partitions of the set {1,2}:
  (1)(2), (1)[2], [1](2), [1][2], (12), [12], [21] where parenthesis denote blocks consisting of cycles, and square brackets denote blocks consisting of linear orders. (End)

			E.g.f.: A(x) = 1 + 2*x + 7*x^2/2! + 26*x^3/3! + 115*x^4/4! + 542*x^5/5! + ...

Column k=3 of A376826.

Cf. A202829.

a := n -> I^(1-n)*2^((1+n)/2)*3^((n-1)/2)*KummerU((1-n)/2, 3/2, -2/3);
seq(round(evalf(a(n), 32)), n=0..23); # Peter Luschny, May 15 2014

CoefficientList[Series[E^(2*x+3*x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := Sum[StirlingS1[n, k]*3^n/2^(n - k) BellB[k, 2/3], {k, 0, n}]; Table[a[n], {n, 0, 12}] (* Emanuele Munarini, May 15 2014 *)

B(n,x) := sum(stirling2(n,k)*x^k,k,0,n);
a(n) := sum(stirling1(n,k)*3^n/2^(n-k)*B(k,2/3),k,0,n);
makelist(a(n),n,0,40); /* Emanuele Munarini, May 15 2014 */

{a(n)=n!*polcoeff(exp(2*x+3*x^2/2+x*O(x^n)),n)}

{a(n)=sum(k=0,n\2,2^(n-3*k)*3^k*n!/((n-2*k)!*k!))}

/* O.g.f. as a continued fraction: */
{a(n)=local(CF=1+2*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-2*x-3*(n-k)*x^2*CF)); polcoeff(CF, n)}

a(n) = Sum_{k=0..floor(n/2)} 2^(n-3*k)*3^k * n!/((n-2*k)!*k!).
O.g.f.: 1/(1-2*x - 3*x^2/(1-2*x - 6*x^2/(1-2*x - 9*x^2/(1-2*x - 12*x^2/(1-2*x -...))))), (continued fraction).
E.g.f.: exp((4*x+3*x^2)/2) = G(0); G(k) = 1+(4*x+3*x^2)/(4*k+2-(4*x+3*x^2)*(4*k+2)/(4*x+3*x^2+4*(k+1)/G(k+1))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 28 2011
a(n) = 2*a(n-1) + 3*(n-1)*a(n-2), a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 29 2012
a(n) ~ exp((2/3)*sqrt(3*n) - n/2 - 1/3)*3^(n/2)*n^(n/2)/sqrt(2)*(1 + (11/54)*sqrt(3)/sqrt(n)). - Vaclav Kotesovec, Oct 20 2012
a(n) = Sum_{k=0..n} s(n,k)*(-1)^(n-k)*(3^n/2^(n-k))*B(k,2/3), where the s(n,k) are the (signless) Stirling numbers of the first kind and the B(n,x) = Sum_{k=0..n} S(n,k)*x^k are the Stirling polynomials (or exponential polynomials), where the S(n,k) are the Stirling numbers of the second kind. - Emanuele Munarini, May 15 2014
a(n) = i^(1-n)*2^((1+n)/2)*3^((n-1)/2)*U((1-n)/2, 3/2, -2/3), U Kummer's hypergeometric function of the second kind. - Peter Luschny, May 15 2014

A202832 E.g.f: exp(2x + 5x^2/2).

Original entry on oeis.org

1, 2, 9, 38, 211, 1182, 7639, 50738, 368841, 2767202, 22132249, 182624598, 1582522891, 14122521662, 131109031239, 1250794578818, 12334766500561, 124733099306562, 1297921351160809, 13821821639912198, 150946171640101251, 1684074507271422302, 19217497036753475959

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: 1 + 2*x + 9*x^2/2! + 38*x^3/3! + 211*x^4/4! + 1182*x^5/5! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=5 of A376826.

Cf. A202831.

CoefficientList[Series[E^(2*x+5*x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n)=n!*polcoeff(exp(2*x+5*x^2/2+x*O(x^n)),n)}

{a(n)=sum(k=0,n\2,2^(n-3*k)*5^k*n!/((n-2*k)!*k!))}

/* O.g.f. as a continued fraction: */
{a(n)=local(CF=1+2*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-2*x-5*(n-k)*x^2*CF)); polcoeff(CF, n)}

a(n) = Sum_{k=0..[n/2]} 2^(n-3*k)*5^k * n!/((n-2*k)!*k!).
O.g.f.: 1/(1-2*x - 5*x^2/(1-2*x - 10*x^2/(1-2*x - 15*x^2/(1-2*x - 20*x^2/(1-2*x -...))))), a continued fraction.
Recurrence: a(n) = 2*a(n-1) + 5*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ exp(2/5*sqrt(5*n)-n/2-1/5)*5^(n/2)*n^(n/2)/sqrt(2)*(1+17/150*sqrt(5)/sqrt(n)). - Vaclav Kotesovec, Oct 20 2012

A373625 Sum of all entries in character table of the hyperoctahedral group B_n.

Original entry on oeis.org

1, 2, 8, 26, 112, 410, 1860, 8074, 40376, 199050, 1085232, 5923394, 34842408, 206403234, 1295653484, 8219293954, 54613967584, 367414298386, 2567777927672, 18187100499306, 133016727225888, 986352813933034, 7518613974827732, 58110359176236314, 460095738657984024

Offset: 0

Arvind Ayyer, Jun 11 2024

nonn

			a(2) = 8 because the character table of B_2 is  [[1  1  1  1  1], [ 1 -1 -1  1  1], [ 1 -1  1 -1  1], [ 1  1 -1 -1  1], [ 2  0  0  0 -2]].

Andrew Howroyd, Table of n, a(n) for n = 0..500
Arvind Ayyer, Hiranya Kishore Dey and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv preprint arXiv:2406.06036, [math.RT], 2024.

Cf. A001147, A376826, A082733.

\\ here B(k,n) is o.g.f. of column k of A376826.
B(k,n)={serlaplace(exp(2*x + k*x^2/2 + O(x*x^n)))}
seq(n)={my(d=serlaplace(1/sqrt(1 - 2*x + O(x*x^(n\2))))); Vec(prod(i=1, (n+1)\2, subst(d + O(x^(n\(2*i)+1)), x, 2*i*x^(2*i))^(2-i%2) * subst(B(4*i-2, n\(2*i-1)), x, x^(2*i-1))))} \\ Andrew Howroyd, Oct 07 2024

G.f.: Product_{i >= 1} D(4*i*x^(4*i)) * D(2*i*x^(2*i)) * R(2*i-1, x^(2*i-1)), where D(x) is the g.f. of A001147, R(r, x) = Sum_{k>=0} c(r,k)*x^k and c(r,n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * 2^(n-k) * (2*k-1)!! * r^k. [edited by Andrew Howroyd, Oct 07 2024]

G.f.: Product_{i >= 1} D(4*i*x^(4*i)) * D(2*i*x^(2*i)) * B(4*i-2, x^(2*i-1)), where D(x) is the g.f. of A001147 and B(k,x) is the g.f. of column k of A376826. - Andrew Howroyd, Oct 07 2024

a(0)=1 prepended and a(10) onwards from Andrew Howroyd, Oct 06 2024

cp's OEIS Frontend

A000898 a(n) = 2*(a(n-1) + (n-1)*a(n-2)) for n >= 2 with a(0) = 1.

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A193778 Number of signed permutations of length 2n invariant under D and D'bar.

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A202830 E.g.f.: exp(2*x + 3*x^2/2).

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Maple

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A202832 E.g.f: exp(2*x + 5*x^2/2).

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Mathematica

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PARI

PARI

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A373625 Sum of all entries in character table of the hyperoctahedral group B_n.

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Examples

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Programs

PARI

Formula

Extensions

A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.

A202830 E.g.f.: exp(2x + 3x^2/2).

A202832 E.g.f: exp(2x + 5x^2/2).