A293720 -id:A293720 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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)

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

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).

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

[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

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

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 *)

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

[(-i)^n*hermite(n,i/2) for n in range(41)] # G. C. Greubel, Jul 12 2024

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

A293604 Expansion of e.g.f.: exp(x * (1 - x)).

Original entry on oeis.org

1, 1, -1, -5, 1, 41, 31, -461, -895, 6481, 22591, -107029, -604031, 1964665, 17669471, -37341149, -567425279, 627491489, 19919950975, -2669742629, -759627879679, -652838174519, 31251532771999, 59976412450835, -1377594095061119, -4256461892701199

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..732

Cf. A000321, A111884, A293571, A293572, A293573.

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), this sequence (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-x^2) ))); // G. C. Greubel, Jul 12 2024

CoefficientList[Series[E^(x*(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)

my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x))))

a(n) = polhermite(n, 1/2); \\ Michel Marcus, Oct 13 2017

[hermite(n, 1/2) for n in range(31)] # G. C. Greubel, Jul 12 2024

a(n) = (-1)^n * A000321(n).
a(n) = a(n-1) - 2 * (n-1) * a(n-2) for n > 1.
E.g.f.: Product_{k>=1} (1 + x^k)^(mu(k)/k). - Ilya Gutkovskiy, May 23 2019
a(n) = Hermite(n, 1/2). - G. C. Greubel, Jul 12 2024

A115329 Expansion of e.g.f.: exp(x + 2*x^2).

Original entry on oeis.org

1, 1, 5, 13, 73, 281, 1741, 8485, 57233, 328753, 2389141, 15539261, 120661465, 866545993, 7140942173, 55667517781, 484124048161, 4046845186145, 36967280461093, 328340133863533, 3137853448906601, 29405064157989241

Offset: 0

Paul D. Hanna, Jan 20 2006

Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.
Combinatorial interpretation: a(n) counts the partitions of a set of n distinguishable objects into subsets of size 1 and 2 with the additional feature that the constituents of the subset of size 2 acquire 2 colors. - Karol A. Penson and P. Blasiak (blasiak(AT)lptl.jussieu.fr), Jun 03 2006
In general, e.g.f. exp(x+m*x^2) has general term sum{k=0..n, C(n,k)*m^k*(n-k)!/(n-m*k)!}. [Paul Barry, Nov 07 2008]
The sequence terms have the form 4*m + 1 (follows from the recurrence). a(n+k) = a(n) (mod k) holds for all n and k by an induction argument making use of the recurrence equation. For each k the sequence a(n) taken modulo k is thus periodic with exact period dividing k. - Peter Bala, Nov 15 2017

Seiichi Manyama, Table of n, a(n) for n = 0..665 (terms 0..200 from Vincenzo Librandi)
Magdalena Boos, Giovanni Cerulli Irelli, Francesco Esposito, Parabolic orbits of 2-nilpotent elements for classical groups, arXiv:1802.06425 [math.RT], 2018.

Column k=4 of A359762.
Cf. A000085, A047974, A115330.
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), A047974 (q=1), this sequence (q=2), A293720 (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x+2*x^2) ))); // G. C. Greubel, Jul 12 2024

a := n -> I^(1 - n)*2^((3*(n - 1))/2)*KummerU((1 - n)/2, 3/2, -1/8):
seq(simplify(a(n)), n=0..21); # Peter Luschny, Nov 21 2017

Range[0, 20]! CoefficientList[Series[Exp[(x + 2 x^2)], {x, 0, 20}], x] (* Vincenzo Librandi, May 22 2013 *)

a(n)=local(m=4);n!*polcoeff(exp(x+m/2*x^2+x*O(x^n)),n)

[(-i*sqrt(2))^n*hermite(n, i/(2*sqrt(2))) for n in range(31)] # G. C. Greubel, Jul 12 2024

Term-by-term square equals A115330 which has e.g.f.: exp(x/(1-4*x))/sqrt(1-16*x^2).
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)2^k*n!/(n-k)! = Sum_{k=0..n} C(n,k)2^k*(n-k)!/(n-2k)!. - Paul Barry, Nov 07 2008
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+8*x)*d/dx. Cf. A000085 and A047974. - Peter Bala, Dec 07 2011
a(n) = a(n-1) + 4*(n-1)*a(n-2). - R. J. Mathar, Dec 10 2011
a(n) ~ 2^(n-1/2)*exp(sqrt(n)/2-n/2-1/16)*n^(n/2). - Vaclav Kotesovec, Oct 19 2012
G.f.: 1/Q(0), where Q(k)= 1 + 4*x*k - x/(1 - 4*x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013
G.f.: 1/G(0), where G(k)= 1 - x - 4*(k+1)*x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 21 2013
a(n) = i^(1 - n)*2^(3*(n - 1)/2)*KummerU((1 - n)/2, 3/2, -1/8). - Peter Luschny, Nov 21 2017
a(n) = (-i*sqrt(2))^n * Hermite(n, i/(2*sqrt(2))). - G. C. Greubel, Jul 12 2024

More terms from Karol A. Penson and P. Blasiak (blasiak(AT)lptl.jussieu.fr), Jun 03 2006

A158954 Numerator of Hermite(n, 1/4).

Original entry on oeis.org

1, 1, -7, -23, 145, 881, -4919, -47207, 228257, 3249505, -13184999, -273145399, 887134513, 27109092817, -65152896535, -3101371292039, 4716976292161, 401692501673153, -239816274060743, -58083536514994775, -21631462857761839, 9271734379541402161

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 1/2, -7/4, -23/8, 145/16, 881/32, -4919/64, -47207/128, 228257/256, 3249505/512, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

Cf. A000079 (denominators).

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), this sequence (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

[Numerator((&+[(-1)^k*Factorial(n)*(1/2)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 09 2018

A158954 := proc(n)
        orthopoly[H](n,1/4) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Numerator[Table[HermiteH[n,1/4],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Mar 23 2011 *)

a(n)=numerator(polhermite(n,1/4)) \\ Charles R Greathouse IV, Jan 29 2016

[2^n*hermite(n, 1/4) for n in range(31)] # G. C. Greubel, Jul 12 2024

D-finite with recurrence a(n) - a(n-1) + 8*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 16 2014
From G. C. Greubel, Jun 09 2018: (Start)
a(n) = 2^n*Hermite(n,1/4).
E.g.f.: exp(x-4*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/2)^(n-2k)/(k!*(n-2k)!)). (End)

A362176 Expansion of e.g.f. exp(x * (1-2*x)).

Original entry on oeis.org

1, 1, -3, -11, 25, 201, -299, -5123, 3249, 167185, 50221, -6637179, -8846903, 309737689, 769776645, -16575533939, -62762132639, 998072039457, 5265897058909, -66595289781995, -466803466259079, 4860819716300521, 44072310882063157, -383679824152382691

Offset: 0

Seiichi Manyama, Apr 10 2023

Seiichi Manyama, Table of n, a(n) for n = 0..668

Column k=4 of A362277.

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), this sequence (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-2*x^2) ))); // G. C. Greubel, Jul 12 2024

With[{m=30}, CoefficientList[Series[Exp[x-2*x^2], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Jul 12 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-2*x))))

[(-sqrt(2))^n*hermite(n, 1/(2*sqrt(2))) for n in range(31)] # G. C. Greubel, Jul 12 2024

a(n) = a(n-1) - 4*(n-1)*a(n-2) for n > 1.
a(n) = n! * Sum_{k=0..floor(n/2)} (-2)^k / (k! * (n-2*k)!).
a(n) = (-sqrt(2))^n * Hermite(n, 1/(2*sqrt(2))). - G. C. Greubel, Jul 12 2024

A362177 Expansion of e.g.f. exp(x * (1-3*x)).

Original entry on oeis.org

1, 1, -5, -17, 73, 481, -1709, -19025, 52753, 965953, -1882709, -59839889, 64418905, 4372890913, -651783677, -367974620369, -309314089439, 35016249465985, 66566286588763, -3715188655737617, -11303745326856599, 434518893361657441, 1858790804545588915

Offset: 0

Seiichi Manyama, Apr 10 2023

Winston de Greef, Table of n, a(n) for n = 0..628

Column k=6 of A362277.

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), this sequence (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-3*x^2) ))); // G. C. Greubel, Jul 12 2024

With[{m=30}, CoefficientList[Series[Exp[x-3*x^2], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Jul 12 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-3*x))))

[(-sqrt(3))^n*hermite(n, 1/(2*sqrt(3))) for n in range(31)] # G. C. Greubel, Jul 12 2024

a(n) = a(n-1) - 6*(n-1)*a(n-2) for n > 1.
a(n) = n! * Sum_{k=0..floor(n/2)} (-3)^k / (k! * (n-2*k)!).
a(n) = (-sqrt(3))^n * Hermite(n, 1/(2*sqrt(3))). - G. C. Greubel, Jul 12 2024

A158968 Numerator of Hermite(n, 1/6).

Original entry on oeis.org

1, 1, -17, -53, 865, 4681, -73169, -578717, 8640577, 91975825, -1307797649, -17863446149, 241080488353, 4099584856537, -52313249418065, -1085408633265389, 13039168709612161, 325636855090044193, -3664348770051277073, -109170689819225595605, 1144036589538311163361

Offset: 0

N. J. A. Sloane, Nov 12 2009

G. C. Greubel, Table of n, a(n) for n = 0..450
DLMF, Digital library of mathematical functions, Table 18.9.1 for H_n(x).

Cf. A158811, A158960.

Sequences with e.g.f = exp(x + q*x^2): this sequence (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

[Numerator((&+[(-1)^k*Factorial(n)*(1/3)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 10 2018

Numerator[Table[HermiteH[n,1/6],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011 *)
Table[3^n*HermiteH[n, 1/6], {n,0, 50}] (* G. C. Greubel, Jul 10 2018 *)

a(n)=numerator(polhermite(n,1/6)) \\ Charles R Greathouse IV, Jan 29 2016

[3^n*hermite(n, 1/6) for n in range(31)] # G. C. Greubel, Jul 12 2024

From G. C. Greubel, Jun 02 2018: (Start)
a(n) = 3^n * Hermite(n, 1/6).
E.g.f.: exp(x - 9*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/3)^(n-2*k)/(k!*(n-2*k)!)). (End)
D-finite with recurrence a(n) -a(n-1) +18*(n-1)*a(n-2)=0. - [DLMF] Georg Fischer, Feb 06 2021

A359762 Array read by ascending antidiagonals. T(n, k) = n![x^n] exp(x + (k/2) x^2). A generalization of the number of involutions (or of 'telephone numbers').

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 10, 7, 4, 1, 1, 1, 26, 25, 10, 5, 1, 1, 1, 76, 81, 46, 13, 6, 1, 1, 1, 232, 331, 166, 73, 16, 7, 1, 1, 1, 764, 1303, 856, 281, 106, 19, 8, 1, 1, 1, 2620, 5937, 3844, 1741, 426, 145, 22, 9, 1, 1

Offset: 0

Peter Luschny, Jan 14 2023

The array is a generalization of the number of involutions of permutations on n letters, A000085, also known as 'telephone numbers'. According to Bednarz et al. the telephone number interpretation "is due to John Riordan, who noticed that T(n, 1) is the number of connection patterns in a telephone system with n subscribers."

In graph theory, the n-th telephone number is the total number of matchings of a complete graph K_n (see the Wikipedia entry). Assuming a network with k possibilities of connections leads to a network that can be modeled by a complete multigraph K(n, k). The total number of connection patterns in such a network is given by T(n, k).

			Array T(n, k) starts:
  [n\k] 0   1      2        3       4        5        6        7
  --------------------------------------------------------------
  [0] 1,    1,     1,       1,      1,       1,       1,       1, ... [A000012]
  [1] 1,    1,     1,       1,      1,       1,       1,       1, ... [A000012]
  [2] 1,    2,     3,       4,      5,       6,       7,       8, ... [A000027]
  [3] 1,    4,     7,      10,     13,      16,      19,      22, ... [A016777]
  [4] 1,   10,    25,      46,     73,     106,     145,     190, ... [A100536]
  [5] 1,   26,    81,     166,    281,     426,     601,     806, ...
  [6] 1,   76,   331,     856,   1741,    3076,    4951,    7456, ...
  [7] 1,  232,  1303,    3844,   8485,   15856,   26587,   41308, ...
  [8] 1,  764,  5937,   21820,  57233,  123516,  234529,  406652, ...
  [9] 1, 2620, 26785,  114076, 328753,  757756, 1510705, 2719900, ...
   [A000085][A047974][A115327][A115329][A115331]

John Riordan, Introduction to Combinatorial Analysis, Dover (2002).

Urszula Bednarz and Małgorzata Wołowiec-Musiał, On a new generalization of telephone numbers, Turkish Journal of Mathematics: Vol. 43: No. 3, (2019).
Carlos M. da Fonseca and Anthony G. Shannon, Telephone numbers extensions, J. Interdisc. Math. (2025) Vol. 28, No. 4, 1573-1580. See pp. 1574, 1577.
Wikipedia, Telephone numbers.

Rows include: A000012, A000027, A016777, A100536.
Columns include: A000012, A000085, A047974, A115327, A115329, A115331, A293720.
Cf. A000085, A277614, A359760.

T := (n, k) -> add(binomial(n, j)*doublefactorial(j-1)*k^(j/2), j = 0..n, 2):
for n from 0 to 9 do lprint(seq(T(n, k), k = 0..7)) od;
T := (n, k) -> ifelse(k=0, 1, I^(-n)*(2*k)^(n/2)*KummerU(-n/2, 1/2, -1/(2*k))):
seq(seq(simplify(T(n-k, k)), k = 0..n), n = 0..10);
T := proc(n, k) exp(x + (k/2)*x^2): series(%, x, 16): n!*coeff(%, x, n) end:
seq(lprint(seq(simplify(T(n, k)), k = 0..8)), n = 0..9);
T := proc(n, k) option remember; if n = 0 or n = 1 then 1 else T(n, k-1) +
n*(k-1)*T(n, k-2) fi end: for n from 0 to 9 do seq(T(n, k), k=0..9) od;
# Only to check the interpretation as a determinant of a lower Hessenberg matrix:
gen := proc(i, j, n) local ev, tv; ev := irem(j+i, 2) = 0; tv := j < i and not ev;
if j > i + 1 then 0 elif j = i + 1 then -1 elif j <= i and ev then 1
elif tv and i < n then x*(n + 1 - i) - 1 else x fi end:
det := M -> LinearAlgebra:-Determinant(M):
p := (n, k) -> subs(x = k, det(Matrix(n, (i, j) -> gen(i, j, n)))):
for n from 0 to 9 do seq(p(n, k), k = 0..7) od;

T[n_, k_] := Sum[Binomial[n, j] Factorial2[j-1] * If[j==0, 1,  k^(j/2)], {j, 0, n, 2}];
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *)

from math import factorial, comb
def oddfactorial(n: int) -> int:
    return factorial(2 * n) // (2**n * factorial(n))
def T(n: int, k: int) -> int:
    return sum(comb(n, 2 * j) * oddfactorial(j) * k**j for j in range(n + 1))
for n in range(10): print([T(n, k) for k in range(8)])

T(n, k) = Sum_{j=0..n, j even} binomial(n, j) * (j - 1)!! * k^(j/2).
T(n, k) = T(n, k-1) + n*(k-1)*T(n, k-2) for n >= 2, T(n, 0) = T(n, 1) = 1.
T(n, k) = i^(-n) * (2*k)^(n/2) * KummerU(-n/2, 1/2, -1/(2*k)) for k >= 1, and T(n, 0) = 1.

A293724 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} j^2*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 9, 25, 1, 1, 1, 9, 79, 241, 1, 1, 1, 9, 79, 457, 1041, 1, 1, 1, 9, 79, 841, 5901, 10681, 1, 1, 1, 9, 79, 841, 7821, 66841, 60649, 1, 1, 1, 9, 79, 841, 10821, 118681, 720259, 658785, 1, 1, 1, 9, 79, 841, 10821, 136681, 1782019

Offset: 0

Seiichi Manyama, Oct 15 2017

			Square array begins:
   1,    1,    1,    1,     1, ...
   1,    1,    1,    1,     1, ...
   1,    9,    9,    9,     9, ...
   1,   25,   79,   79,    79, ...
   1,  241,  457,  841,   841, ...
   1, 1041, 5901, 7821, 10821, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..4 give A000012, A293720, A293721, A293723.
Rows n=0-1 give A000012.
Main diagonal gives A255807.
Cf. A293669, A293718.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k,n)} j^3*A(n-j,k)/(n-j)!.

cp's OEIS Frontend

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

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A293604 Expansion of e.g.f.: exp(x * (1 - x)).

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A115329 Expansion of e.g.f.: exp(x + 2*x^2).

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A158954 Numerator of Hermite(n, 1/4).

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A362176 Expansion of e.g.f. exp(x * (1-2*x)).

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A362177 Expansion of e.g.f. exp(x * (1-3*x)).

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

A359762 Array read by ascending antidiagonals. T(n, k) = n![x^n] exp(x + (k/2) x^2). A generalization of the number of involutions (or of 'telephone numbers').