A093883 Product of all possible sums of two distinct numbers taken from among first n natural numbers.
1, 3, 60, 12600, 38102400, 2112397056000, 2609908810629120000, 84645606509847871488000000, 82967862872337478796810649600000000, 2781259372192376861719959017613164544000000000
Offset: 1
Keywords
A001861 Expansion of e.g.f. exp(2*(exp(x) - 1)).
1, 2, 6, 22, 94, 454, 2430, 14214, 89918, 610182, 4412798, 33827974, 273646526, 2326980998, 20732504062, 192982729350, 1871953992254, 18880288847750, 197601208474238, 2142184050841734, 24016181943732414, 278028611833689478, 3319156078802044158, 40811417293301014150
Offset: 0
Comments
Values of Bell polynomials: ways of placing n labeled balls into n unlabeled (but 2-colored) boxes.
First column of the square of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms, Mar 30 2007
Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939. - Gottfried Helms, Apr 08 2007
Equals row sums of triangle A144061. - Gary W. Adamson, Sep 09 2008
Equals eigensequence of triangle A109128. - Gary W. Adamson, Apr 17 2009
Hankel transform is A108400. - Paul Barry, Apr 29 2009
The number of ways of putting n labeled balls into a set of bags and then putting the bags into 2 labeled boxes. An example is given below. - Peter Bala, Mar 23 2013
The f-vectors of n-dimensional hypercube are given by A038207 = exp[M*B(.,2)] = exp[M*A001861(.)] where M = A238385-I and (B(.,x))^n = B(n,x) are the Bell polynomials (cf. A008277). - Tom Copeland, Apr 17 2014
Moments of the Poisson distribution with mean 2. - Vladimir Reshetnikov, May 17 2016
Exponential self-convolution of Bell numbers (A000110). - Vladimir Reshetnikov, Oct 06 2016
Examples
a(2) = 6: The six ways of putting 2 balls into bags (denoted by { }) and then into 2 labeled boxes (denoted by [ ]) are
01: [{1,2}] [ ];
02: [ ] [{1,2}];
03: [{1}] [{2}];
04: [{2}] [{1}];
05: [{1} {2}] [ ];
06: [ ] [{1} {2}].
- _Peter Bala_, Mar 23 2013
References
- 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
- Seiichi Manyama, Table of n, a(n) for n = 0..558 (terms 0..100 from T. D. Noe)
- M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
- Michael Anshelevich, Product formulas on posets, Wick products, and a correction for the q-Poisson process, arXiv:1708.08034 [math.OA], 2017, See Proposition 34 p. 25.
- Diego Arcis, Camilo González, and Sebastián Márquez, Symmetric functions in noncommuting variables in superspace, arXiv:2312.00574 [math.CO], 2023.
- 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.
- J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
- J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
- Jacques Carlier and Corinne Lucet, A decomposition algorithm for network reliability evaluation. In First International Colloquium on Graphs and Optimization (GOI), 1992 (Grimentz). Discrete Appl. Math. 65 (1996), 141-156 (see page 152 and Fig 6).
- 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
- Wan-Ming Guo and Lily Li Liu, Asymptotic normality of the Stirling-Whitney-Riordan triangle, Filomat (2023) Vol. 37, No. 9, 2923-2934.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 66 [broken link?]
- Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
- G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.
- Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 20.
- T. Mansour, M. Shattuck and D. G. L. Wang, Recurrence relations for patterns of type (2, 1) in flattened permutations, arXiv preprint arXiv:1306.3355 [math.CO], 2013.
- Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
- T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
- OEIS Wiki, Sorting numbers
- J. Riordan, Letter to N. J. A. Sloane, Oct. 1970
- J. Riordan, Letter, Oct 31 1977
- Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. See Table 5.1. - From _N. J. A. Sloane_, Jan 04 2013
- Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 24 2012
- Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Crossrefs
Programs
-
Magma
[&+[2^k*StirlingSecond(n, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, May 18 2019
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Maple
A001861:=n->add(Stirling2(n,k)*2^k, k=0..n); seq(A001861(n), n=0..20); # Wesley Ivan Hurt, Apr 18 2014 # second Maple program: b:= proc(n, m) option remember; `if`(n=0, 2^m, m*b(n-1, m)+b(n-1, m+1)) end: a:= n-> b(n, 0): seq(a(n), n=0..25); # Alois P. Heinz, Aug 04 2021
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Mathematica
Table[Sum[StirlingS2[n, k]*2^k, {k, 0, n}], {n, 0, 21}] (* Geoffrey Critzer, Oct 06 2009 *) mx = 16; p = 1; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *) Table[BellB[n, 2], {n, 0, 20}] (* Vaclav Kotesovec, Jan 06 2013 *) -
PARI
a(n)=if(n<0,0,n!*polcoeff(exp(2*(exp(x+x*O(x^n))-1)),n))
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PARI
{a(n)=polcoeff(sum(m=0, n, 2^m*x^m/prod(k=1,m,1-k*x +x*O(x^n))), n)} /* Paul D. Hanna, Feb 15 2012 */ -
PARI
{a(n) = sum(k=0, n, 2^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 28 2019 -
Sage
expnums(30, 2) # Zerinvary Lajos, Jun 26 2008
Formula
a(n) = Sum_{k=0..n} 2^k*Stirling2(n, k). - Emeric Deutsch, Oct 20 2001
a(n) = exp(-2)*Sum_{k>=1} 2^k*k^n/k!. - Benoit Cloitre, Sep 25 2003
G.f. satisfies 2*(x/(1-x))*A(x/(1-x)) = A(x) - 1; twice the binomial transform equals the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
PE = exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,1]. - Gottfried Helms, Apr 08 2007
G.f.: 1/(1-2x-2x^2/(1-3x-4x^2/(1-4x-6x^2/(1-5x-8x^2/(1-6x-10x^2/(1-... (continued fraction). - Paul Barry, Apr 29 2009
O.g.f.: Sum_{n>=0} 2^n*x^n / Product_{k=1..n} (1-k*x). - Paul D. Hanna, Feb 15 2012
a(n) ~ exp(-2-n+n/LambertW(n/2))*n^n/LambertW(n/2)^(n+1/2). - Vaclav Kotesovec, Jan 06 2013
G.f.: (G(0) - 1)/(x-1)/2 where G(k) = 1 - 2/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: 1/Q(0) where Q(k) = 1 + x*k - x - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
G.f.: ((1+x)/Q(0)-1)/(2*x), where Q(k) = 1 - (k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: T(0)/(1-2*x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-2*x-x*k)*(1-3*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013
a(n) = Sum_{k=0..n} A033306(n,k) = Sum_{k=0..n} binomial(n,k)*Bell(k)*Bell(n-k), where Bell = A000110 (see Motzkin, p. 170). - Danny Rorabaugh, Oct 18 2015
a(0) = 1 and a(n) = 2 * Sum_{k=0..n-1} binomial(n-1,k)*a(k) for n > 0. - Seiichi Manyama, Sep 25 2017 [corrected by Ilya Gutkovskiy, Jul 12 2020]
A047974 a(n) = a(n-1) + 2*(n-1)*a(n-2).
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
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
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Peter Alspaugh, James Garrett, Nataša Jonoska, and Masahico Saito, Structures of Monoids Motivated by DNA Origami, arXiv:2501.14966 [math.RA], 2025. See p. 22.
- Tewodros Amdeberhan, Valerio De Angelis, Atul Dixit, Victor H. Moll, and Christophe Vignat, From sequences to polynomials and back, via operator orderings, J. Math. Phys. 54, 123502 (2013); Alternative copy
- Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
- Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts); Alternative copy
- Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394; Alternative copy.
- Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011.
- Samuele Giraudo, Combalgebraic structures on decorated cliques, arXiv:1709.08416 [math.CO], 2017; and also, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8.
- Tom Halverson and Mike Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150 [math.RT], 2013.
- Andrei Khruzin, Enumeration of chord diagrams, arXiv:math/0008209 [math.CO], 2000.
- G. Latouche and P. G. Taylor, A stochastic fluid model for an ad hoc mobile network, Queueing Syst. 63, No. 1-4, 109-129 (2009), eq. (1).
- Robert A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
- Jocelyn Quaintance and Harris Kwong, Permutations and combinations of colored multisets, JIS 13 (2010) #10.2.6.
- Index entries for related partition-counting sequences
- Index entries for sequences related to Hermite polynomials
Programs
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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
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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
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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
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
Comments
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
Examples
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
References
- 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).
Links
- 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
Crossrefs
Programs
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Haskell
a000898 n = a000898_list !! n a000898_list = 1 : 2 : (map (* 2) $ zipWith (+) (tail a000898_list) (zipWith (*) [1..] a000898_list)) -- Reinhard Zumkeller, Oct 10 2011
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Maple
# For Maple program see A000903. seq(simplify((-I)^n*HermiteH(n, I)), n=0..25); # Peter Luschny, Oct 23 2015
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Mathematica
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 *) -
Maxima
makelist((%i)^n*hermite(n,-%i),n,0,12); /* Emanuele Munarini, Mar 02 2016 */
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PARI
{a(n) = if( n<0, 0, n! * polcoeff( exp(2*x + x^2 + x * O(x^n)), n))}; /* Michael Somos, Feb 08 2004 */ -
PARI
{a(n) = if( n<2, max(0, n+1), 2*a(n-1) + (2*n - 2) * a(n-2))}; /* Michael Somos, Feb 08 2004 */ -
PARI
my(x='x+O('x^66)); Vec(serlaplace(exp(2*x+x^2))) \\ Joerg Arndt, Oct 04 2013 -
PARI
{a(n) = sum(k=0, n\2, 2^(n - 2*k) * n! / (k! * (n - 2*k)!))}; /* Michael Somos, Oct 23 2015 */
Formula
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
Extensions
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
A009235 E.g.f. exp( sinh(x) / exp(x) ) = exp( (1-exp(-2*x))/2 ).
1, 1, -1, -1, 9, -23, -25, 583, -3087, 4401, 79087, -902097, 4783801, 2361049, -348382697, 4102879415, -24288551071, -47413121055, 3214104039007, -44472852461857, 326386562502889, 417716032223049, -55104307651136313, 962111031220099495
Offset: 0
Comments
Hankel transform is (-1)^binomial(n+1,2)*A108400. - Paul Barry, Apr 15 2010
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..529
- I. M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121 [math.CO], 2001.
Programs
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Maple
a := n -> (-2)^n*add(Stirling2(n,k)*(-1/2)^k, k=0..n): seq(a(n), n=0..23); # Peter Luschny, Jan 06 2020
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Mathematica
With[{nn=30},CoefficientList[Series[Exp[Sinh[x]/Exp[x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Jan 07 2013 *) Table[(-2)^n BellB[n, -1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *) -
PARI
x='x+O('x^66); /* that many terms */ v=Vec(serlaplace(exp(sinh(x)/exp(x)))) /* Joerg Arndt, May 19 2012 */
Formula
a(n) = Sum_{k=0..n} (-2)^(n-k)*Stirling2(n, k). - Vladeta Jovovic, Apr 04 2003
From Peter Bala, May 16 2012: (Start)
Recurrence equation: a(n+1) = Sum_{k = 0..n} (-2)^(n-k)*C(n,k)*a(k). Written umbrally this is a(n+1) = (a-2)^n (expand the binomial and replace a^k with a(k)). More generally, a*f(a) = f(a-2) holds umbrally for any polynomial f(x). An inductive argument then establishes the umbral recurrence a*(a+2)*(a+4)*...*(a+2*(n-1)) = 1 with a(0) = 1. Cf. A004211.
Touchard's congruence holds for odd prime p: a(p+k) = (a(k) + a(k+1)) (mod p) for k = 0,1,2, ... (adapt the proof of Theorem 10.1 in Gessel). In particular, a(p) = 2 (mod p) for odd prime p. (End)
From Sergei N. Gladkovskii, Sep 21 2012 - Oct 24 2013: (Start)
Continued fractions:
G.f.: (1/E(0)-1)/x where E(k)= 1 - x/(1 - 2*x + 2*x*(k+1)/E(k+1));
G.f.: 1 +x/G(0) where G(k)= 1 + 2*x/(1 + 1/(1 + 4*x*(k+1)/G(k+1)));
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+x*2*k)/(1-x/(x-1/G(k+1)));
G.f.: 1/Q(0) where Q(k)= 1 - x/(1 + 2*x*(k+1)/Q(k+1) );
G.f.: Q(0)/(1-x), where Q(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) + (1-x+2*x*k)*(1+x+2*x*k)/Q(k+1)). (End)
Lim sup n->infinity (abs(a(n))/n!)^(1/n) / (2*abs(exp(1/LambertW(-2*n)) / LambertW(-2*n))) = 1. - Vaclav Kotesovec, Aug 04 2014
a(n) = (-2)^n*B_n(-1/2), where B_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Oct 20 2015
G.f. A(x) satisfies: A(x) = 1 + x*A(x/(1 + 2*x))/(1 + 2*x). - Ilya Gutkovskiy, May 02 2019
Extensions
Extended with signs by Olivier Gérard, Mar 15 1997
A083886 Expansion of e.g.f. exp(3*x)*exp(x^2).
1, 3, 11, 45, 201, 963, 4899, 26253, 147345, 862083, 5238459, 32957037, 214117209, 1433320515, 9867008979, 69734001357, 505212273441, 3747124863747, 28418591888235, 220152270759597, 1740363304031721, 14027180742479043, 115176800996769411, 962726355659386125, 8186311912829551281, 70769800810139187843
Offset: 0
Comments
Binomial transform of A000898.
Hankel transform is A108400. - Paul Barry, Jun 13 2009
a(n) is the number of self-inverse signed permutations of length 2n that are equal to their reverse-complements and avoid the pattern (-2,-1). As a result, a(n) also gives the same thing but for avoiding any one of (-1,-2), (+2,+1) or (+1,+2) instead of (-2,-1) (See the Hardt and Troyka reference). - Justin M. Troyka, Aug 05 2011
a(n) is also the number of skew-symmetric (n,n)-clans, or the number of B-orbits in the symmetric space of type CI, Sp_{2n}(C)/GL_n(C) where B is a Borel subgroup of Sp_{2n}(C). - Aram Bingham, Oct 08 2019
Examples
Since a(2) = 11, there are 11 self-inverse signed permutations of 4 that are equal to their reverse-complements and avoid (-2,-1). Some of these are: (+3,+4,+1,+2), (+4,-2,-3,+1), (-1,+3,+2,-4), (-1,-2,-3,-4). - _Justin M. Troyka_, Aug 05 2011
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..717
- Aram Bingham and Ozlem Ugurlu, Sects and lattice paths over the Lagrangian Grassmannian, arXiv:1903.07229 [math.CO], 2019.
- A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179--217.
- A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).
Programs
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Mathematica
a = {1, 3}; For[n = 2, n < 26, n++, a = Append[a, 3 a[[n]] + 2 (n - 1) a[[n - 1]]]]; a (* Justin M. Troyka, Aug 05 2011 *) CoefficientList[Series[Exp[3*x+x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *) Table[Abs[HermiteH[n, 3 I/2]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 11 2016 *) -
PARI
my(x='x+O('x^26)); Vec(serlaplace(exp(3*x)*exp(x^2))) /* Joerg Arndt, Jul 12 2012 */
Formula
E.g.f.: exp(3*x+x^2).
From Paul Barry, Jun 13 2009: (Start)
G.f.: 1/(1-3x-2x^2/(1-3x-4x^2/(1-3x-6x^2/(1-3x-8x^2/(1-... (continued fraction);
a(n) = Sum_{k=0..floor(n/2)} C(n,2k) * (2k)! * 3^(n-2k) / k!. (End)
a(n) = i^n*Hermite_H(n, -3i/2), i=sqrt(-1). - Paul Barry, Jun 15 2009
a(0) = 1; a(1) = 3; a(n) = 3*a(n-1) + 2*(n-1)*a(n-2) for n >= 2. - Justin M. Troyka, Aug 05 2011
E.g.f. 1 + (x+3)*x/(G(0)-x^2-3*x) where G(k)= x^2 + 3*x + k + 1 - (x+3)*x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 12 2012
G.f.: 1/Q(0) where Q(k) = 1 + 2*x*k - 2*x - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
a(n) ~ n^(n/2)*2^(n/2-1/2)*exp(3*sqrt(n/2)-n/2-9/8) * (1+21*sqrt(2)/(32*sqrt(n))). - Vaclav Kotesovec, Jun 25 2013
A186002 Hankel transform of A186001.
1, 1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..39
Programs
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Mathematica
Table[2^(1/2*(n - 1)*n)*BarnesG[n + 1], {n, 0, 25}] (* G. C. Greubel, Feb 22 2017 *)
Formula
a(n) = Product_{k=0..n} (2*k+0^k)^(n-k).
Essentially the same as A108400.
From Alexander R. Povolotsky, Feb 10 2011: (Start)
WolframAlpha shows that
a(n) = (0^n*2^(1/2*(n-1)*n)*exp^(1/12-zeta^(1, 0)(-1, n+1)))/A
where zeta(s, a) is the generalized Riemann zeta function and A is the Glaisher-Kinkelin constant.
WolframAlpha suggests that for all terms given
a(n) = 2^(1/2*(n-1)*n)*G(n+1)
where G(n) is the Barnes G-function. (End)
a(n) ~ 2^(n^2/2) * n^(n^2/2 - 1/12) * Pi^(n/2) / (A * exp(3*n^2/4 - 1/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Feb 24 2019
A171151 Expansion of (A(x)-1)/(x*A(x)), A(x) the g.f. of A004211.
1, 2, 6, 26, 142, 898, 6342, 49114, 412046, 3711746, 35660166, 363484058, 3914162830, 44370673282, 527868672582, 6572992645978, 85461951576974, 1157778354181634, 16310949128381190, 238543136307926810
Offset: 0
Comments
Hankel transform is A108400.
Formula
G.f.: 1/(1-2x/(1-x/(1-4x/(1-x/(1-6x/(1-x/(1-8x/(1-x/(1-... (continued function).
a(n) = Sum_{k=0..n} A086329(n,k)*2^k. - Philippe Deléham, Dec 05 2009
G.f.: 1/U(0) where U(k) = 1 - x - 2*x*k + x*(2*x*k + 2*x - 1)/U(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Sep 24 2012
G.f.: 1/x - 1/( x*G(0) - 1 ) where G(k) = 1 - (4*x*k-1)/(x - x^4/(x^3 - (4*x*k-1)*(4*x*k+2*x-1)*(4*x*k+4*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 08 2013
G.f.: (1 - G(0))/x where G(k) = 1 - x/(1 - 2*x*(k + 1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 30 2013
G.f.: 1/x + 1 - (2*x+1)/(G(0) + 2*x+1), where G(k)= 2*x*k - x - 1 - 2*(k+1)*x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 21 2013
G.f.: 1/x - Q(0)/x, where Q(k) = 1 - x*(2*k+1) - (2*k+2)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 09 2013
Comments
Examples
References
Links
Crossrefs
Programs
Maple
Mathematica
f[n_] := Product[(j + k), {k, 2, n}, {j, 1, k - 1}]; Array[f, 10] (* Robert G. Wilson v, Jan 08 2013 *)PARI
Formula
Extensions