A001879 -id:A001879 - cp's OEIS frontend

A001498 Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).

1, 1, 1, 1, 3, 3, 1, 6, 15, 15, 1, 10, 45, 105, 105, 1, 15, 105, 420, 945, 945, 1, 21, 210, 1260, 4725, 10395, 10395, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 1, 36, 630, 6930, 51975, 270270, 945945, 2027025, 2027025, 1, 45, 990, 13860, 135135, 945945, 4729725, 16216200, 34459425, 34459425

Offset: 0

N. J. A. Sloane

The row polynomials with exponents in increasing order (e.g., third row: 1+3x+3x^2) are Grosswald's y_{n}(x) polynomials, p. 18, Eq. (7).
Also called Bessel numbers of first kind.
The triangle a(n,k) has factorization [C(n,k)][C(k,n-k)]Diag((2n-1)!!) The triangle a(n-k,k) is A100861, which gives coefficients of scaled Hermite polynomials. - Paul Barry, May 21 2005
Related to k-matchings of the complete graph K_n by a(n,k)=A100861(n+k,k). Related to the Morgan-Voyce polynomials by a(n,k)=(2k-1)!!*A085478(n,k). - Paul Barry, Aug 17 2005
Related to Hermite polynomials by a(n,k)=(-1)^k*A060821(n+k, n-k)/2^n. - Paul Barry, Aug 28 2005
The row polynomials, the Bessel polynomials y(n,x):=Sum_{m=0..n} (a(n,m)*x^m) (called y_{n}(x) in the Grosswald reference) satisfy (x^2)*(d^2/dx^2)y(n,x) + 2*(x+1)*(d/dx)y(n,x) - n*(n+1)*y(n,x) = 0.
a(n-1, m-1), n >= m >= 1, enumerates unordered n-vertex forests composed of m plane (aka ordered) increasing (rooted) trees. Proof from the e.g.f. of the first column Y(z):=1-sqrt(1-2*z) (offset 1) and the Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w). See their remark on p. 28 on plane recursive trees. For m=1 see the D. Callan comment on A001147 from Oct 26 2006. - Wolfdieter Lang, Sep 14 2007
The asymptotic expansions of the higher order exponential integrals E(x,m,n), see A163931 for information, lead to the Bessel numbers of the first kind in an intriguing way. For the first four values of m these asymptotic expansions lead to the triangles A130534 (m=1), A028421 (m=2), A163932 (m=3) and A163934 (m=4). The o.g.f.s. of the right hand columns of these triangles in their turn lead to the triangles A163936 (m=1), A163937 (m=2), A163938 (m=3) and A163939 (m=4). The row sums of these four triangles lead to A001147, A001147 (minus a(0)), A001879 and A000457 which are the first four right hand columns of A001498. We checked this phenomenon for a few more values of m and found that this pattern persists: m = 5 leads to A001880, m=6 to A001881, m=7 to A038121 and m=8 to A130563 which are the next four right hand columns of A001498. So one by one all columns of the triangle of coefficients of Bessel polynomials appear. - Johannes W. Meijer, Oct 07 2009
a(n,k) also appear as coefficients of (n+1)st degree of the differential operator D:=1/t d/dt, namely D^{n+1}= Sum_{k=0..n} a(n,k) (-1)^{n-k} t^{1-(n+k)} (d^{n+1-k}/dt^{n+1-k}. - Leonid Bedratyuk, Aug 06 2010
a(n-1,k) are the coefficients when expanding (xI)^n in terms of powers of I. Let I(f)(x) := Integral_{a..x} f(t) dt, and (xI)^n := x Integral_{a..x} [ x_{n-1} Integral_{a..x_{n-1}} [ x_{n-2} Integral_{a..x_{n-2}} ... [ x_1 Integral_{a..x_1} f(t) dt ] dx_1 ] .. dx_{n-2} ] dx_{n-1}. Then: (xI)^n = Sum_{k=0..n-1} (-1)^k * a(n-1,k) * x^(n-k) * I^(n+k)(f)(x) where I^(n) denotes iterated integration. - Abdelhay Benmoussa, Apr 11 2025

			The triangle a(n, k), n >= 0, k = 0..n, begins:
  1
  1  1
  1  3   3
  1  6  15    15
  1 10  45   105    105
  1 15 105   420    945    945
  1 21 210  1260   4725  10395   10395
  1 28 378  3150  17325  62370  135135   135135
  1 36 630  6930  51975 270270  945945  2027025  2027025
  1 45 990 13860 135135 945945 4729725 16216200 34459425 34459425
  ...
And the first few Bessel polynomials are:
  y_0(x) = 1,
  y_1(x) = x + 1,
  y_2(x) = 3*x^2 + 3*x + 1,
  y_3(x) = 15*x^3 + 15*x^2 + 6*x + 1,
  y_4(x) = 105*x^4 + 105*x^3 + 45*x^2 + 10*x + 1,
  y_5(x) = 945*x^5 + 945*x^4 + 420*x^3 + 105*x^2 + 15*x + 1,
  ...
Tree counting: a(2,1)=3 for the unordered forest of m=2 plane increasing trees with n=3 vertices, namely one tree with one vertex (root) and another tree with two vertices (a root and a leaf), labeled increasingly as (1, 23), (2,13) and (3,12). - _Wolfdieter Lang_, Sep 14 2007

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

T. D. Noe, Rows n=0..50 of triangle, flattened
Alexander Alldridge, Joachim Hilgert, and Martin R. Zirnbauer, Chevalley's restriction theorem for reductive symmetric superpairs, arXiv:0812.3530 [math.RT], 2008-2009; J. Alg. 323 (4) (2010) 1159-1185 doi:10.1016/j.jalgebra.2009.11.014, Remark 3.17.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
Juan Antonio Barcelo and Anthony Carbery, On the magnitudes of compact sets in Euclidean spaces, arXiv preprint arXiv:1507.02502 [math.MG], 2015.
François Bergeron, Philippe Flajolet, and Bruno Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.
Alexander W. Boldyreff, Decomposition of Rational Fractions into Partial Fractions, Nat. Math. Mag. 17 (6) (1943), 261-267; coefficients (m)N(r).
Alexander Burstein and Toufik Mansour, Words restricted by patterns with at most 2 distinct letters, arXiv:math/0110056 [math.CO], 2001.
Roudy El Haddad, Repeated Integration and Explicit Formula for the n-th Integral of x^m*(ln x)^m', arXiv:2102.11723 [math.GM], 2021.
Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 9.
Emil Grosswald, Bessel Polynomials: Recurrence Relations, Lecture Notes Math. vol. 698, 1978, p. 18.
Cameron Jakub and Mihai Nica, Depth Degeneracy in Neural Networks: Vanishing Angles in Fully Connected ReLU Networks on Initialization, arXiv:2302.09712 [stat.ML], 2023.
Taekyun Kim, and Dae San Kim, Identities involving Bessel polynomials arising from linear differential equations, arXiv:1602.04106 [math.NT], 2016.
H. L. Krall and Orrin Frink, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1949.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First ten rows.
B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
Shi-Mei Ma, Toufik Mansour, and Matthias Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Shi-Mei Ma, Toufik Mansour, Jean Yeh, and Yeong-Nan Yeh, Normal ordered grammars, arXiv:2404.15119 [math.CO], 2024. See p. 11.
Guillermo Navas-Palencia, On the computation of the cumulative distribution function of the Normal Inverse Gaussian distribution, arXiv:2502.16015 [math.NA], 2025. See p. 25.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
John Riordan, Notes to N. J. A. Sloane, Jul. 1968
Florian Stober, Average case considerations for MergeInsertion, Master's Thesis, University of Stuttgart, Institute of Formal Methods in Computer Science, 2018.
Florian Stober and Armin Weiß, On the Average Case of MergeInsertion, arXiv:1905.09656 [cs.DS], 2019.
Laszlo A. Székely, Pál L. Erdős, and M. A. Steel, The combinatorics of evolutionary trees, Séminaire Lotharingien de Combinatoire, B28e (1992), 15 pp.
Juan G. Triana, Bessel polynomials by context-free grammars (Polinomios de Bessel mediante gramáticas independientes del contexto), Bistua, Univ. de Pamplona (Colombia, 2024) Vol 22, No. 2. See p. 3.
Jonas Wahl, Traces on diagram algebras II: Centralizer algebras of easy groups and new variations of the Young graph, arXiv:2009.08181 [math.RT], 2020.
Eric Weisstein's World of Mathematics, Modified Spherical Bessel Function of the Second Kind
Index entries for sequences related to Bessel functions or polynomials

Cf. A001497 (same triangle but rows read in reverse order). Other versions of this same triangle are given in A144331, A144299, A111924 and A100861.
Columns from left edge include A000217, A050534.
Columns 1-6 from right edge are A001147, A001879, A000457, A001880, A001881, A038121.
Bessel polynomials evaluated at certain x are A001515 (x=1, row sums), A000806 (x=-1), A001517 (x=2), A002119 (x=-2), A001518 (x=3), A065923 (x=-3), A065919 (x=4). Cf. A043301, A003215.
Cf. A245066 (central terms). A113025 (y_n(2*x)).

a001498 n k = a001498_tabl !! n !! k
a001498_row n = a001498_tabl !! n
a001498_tabl = map reverse a001497_tabl
-- Reinhard Zumkeller, Jul 11 2014

/* As triangle: */ [[Factorial(n+k)/(2^k*Factorial(n-k)*Factorial(k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 15 2016

Bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end; # explicit Bessel polynomials
Bessel := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*Bessel(n-1)+Bessel(n-2); fi; end; # recurrence for Bessel polynomials
bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;
f := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*f(n-1)+f(n-2); fi; end;
# Alternative:
T := (n,k) -> pochhammer(n+1,k)*binomial(n,k)/2^k:
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, May 11 2018
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (n - k + 1)* T(n, k - 1) + T(n - 1, k) fi fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Oct 02 2023

max=50; Flatten[Table[(n+k)!/(2^k*(n-k)!*k!), {n, 0, Sqrt[2 max]//Ceiling}, {k, 0, n}]][[1 ;; max]] (* Jean-François Alcover, Mar 20 2011 *)

{T(n,k)=if(k<0||k>n, 0, binomial(n, k)*(n+k)!/2^k/n!)} /* Michael Somos, Oct 03 2006 */

A001497_ser(N,t='t) = {
  my(x='x+O('x^(N+2)));
  serlaplace(deriv(exp((1-sqrt(1-2*t*x))/t),'x));
};
concat(apply(Vecrev, Vec(A001497_ser(9)))) \\ Gheorghe Coserea, Dec 27 2017

a(n, k) = (n+k)!/(2^k*(n-k)!*k!) (see Grosswald and Riordan). - Ralf Stephan, Apr 20 2004
a(n, 0)=1; a(0, k)=0, k > 0; a(n, k) = a(n-1, k) + (n-k+1) * a(n, k-1) = a(n-1, k) + (n+k-1) * a(n-1, k-1). - Len Smiley
a(n, m) = A001497(n, n-m) = A001147(m)*binomial(n+m, 2*m) for n >= m >= 0, otherwise 0.
G.f. for m-th column: (A001147(m)*x^m)/(1-x)^(2*m+1), m >= 0, where A001147(m) = double factorials (from explicit a(n, m) form).
Row polynomials y_n(x) are given by D^(n+1)(exp(t)) evaluated at t = 0, where D is the operator 1/(1-t*x)*d/dt. - Peter Bala, Nov 25 2011
G.f.: conjecture: T(0)/(1-x), where T(k) = 1 - x*y*(k+1)/(x*y*(k+1) - (1-x)^2/T(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 13 2013
Recurrence from Grosswald, p. 18, eq. (5), for the row polynomials: y_n(x) = (2*n-1)*x*y_{n-1} + y_{n-2}(x), y_{-1}(x) = 1 = y_{0} = 1, n >= 1. This becomes, for n >= 0, k = 0..n: a(n, k) = 0 for n < k (zeros not shown in the triangle), a(n, -1) = 0, a(0, 0) = 1 = a(1, 0) and otherwise a(n, k) = (2*n-1)*a(n-1, k-1) + a(n-2, k). Compare with the above given recurrences. - Wolfdieter Lang, May 11 2018
T(n, k) = Pochhammer(n+1,k)*binomial(n,k)/2^k = A113025(n,k)/2^k. - Peter Luschny, May 11 2018
a(n, k) = Sum_{i=0..min(n-1, k)} (n-i)(k-i) * a(n-1, i) where x(n) = x*(x-1)*...*(x-n+1) is the falling factorial, this equality follows directly from the operational formula we wrote in Apr 11 2025.- Abdelhay Benmoussa, May 18 2025

A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

Original entry on oeis.org

1, 3, 3, 11, 18, 6, 50, 105, 60, 10, 274, 675, 510, 150, 15, 1764, 4872, 4410, 1750, 315, 21, 13068, 39396, 40614, 19600, 4830, 588, 28, 109584, 354372, 403704, 224490, 68040, 11466, 1008, 36, 1026576, 3518100, 4342080, 2693250, 949095, 198450

Offset: 1

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Oct 22 2009

The higher order exponential integrals E(x,m,n) are defined in A163931. The general formula for the asymptotic expansion E(x,m,n) ~ E(x,m-1,n+1)/x - n*E(x,m-1,n+2)/x^2 + n*(n+1) * E(x,m-1,n+3)/x^3 - n*(n+1)*(n+2)*E(x,m-1,n+4)/x^4 + ...., m >= 1 and n >= 1.
We used this formula and the asymptotic expansion of E(x,m=2,n), see A028421, to determine that E (x,m=3,n) ~ (exp(-x)/x^3)*(1 - (3+3*n)/x + (11+18*n+6*n^2)/x^2 - (50+105*n+ 60*n^2+ 10*n^3)/x^3 + .. ). This formula leads to the triangle coefficients given above.
The asymptotic expansion leads for the values of n from one to ten to known sequences, see the cross-references.
The numerators of the o.g.f.s. of the right hand columns of this triangle lead for z=1 to A001879, see A163938 for more information.
The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=3,n).

			The first few rows of the triangle are:
[1]
[3, 3]
[11, 18, 6]
[50, 105, 60, 10]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A163931 (E(x,m,n)) and A163938.
Cf. A048994 (Stirling1), A000399 (row sums).
A000254, 3*A000399, 6*A000454, 10*A000482, 15*A001233, 21*A001234 equal the first six left hand columns.
A000217, A006011 and A163933 equal the first three right hand columns.
The asymptotic expansion leads to A000399 (n=1), A001706 (n=2), A001712 (n=3), A001717 (n=4), A001722 (n=5), A051525 (n=6), A051546 (n=7), A051561 (n=8), A051563 (n=9) and A051565 (n=10).
Cf. A130534 (m=1), A028421 (m=2) and A163934 (m=4).

nmax:=8; with(combinat): for n1 from 1 to nmax do for m from 1 to n1 do a(n1, m) := (-1)^(n1+m)*binomial(m+1, 2)*stirling1(n1+1, m+1) od: od: seq(seq(a(n1,m), m=1..n1), n1=1..nmax);
# End program 1
with(combinat): imax:=6; EA:=proc(x, m, n) local E, i; E := 0: for i from m-1 to imax+1 do E := E + sum((-1)^(m+k1+1)*binomial(k1, m-1)*n^(k1-m+1)* stirling1(i, k1), k1=m-1..i)/x^(i-m+1) od: E := exp(-x)/x^(m)*E: return(E); end: EA(x, 3, n);
# End program 2

a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+1, 2] * StirlingS1[n+1, m+1]; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 42]] (* Jean-François Alcover, Jun 01 2011, after formula *)

for(n=1,10, for(m=1,n, print1((-1)^(n+m)*binomial(m+1,2) *stirling(n+1,m+1,1), ", "))) \\ G. C. Greubel, Aug 08 2017

a(n,m) = (-1)^(n+m)*binomial(m+1,2)*stirling1(n+1,m+1) for n >= 1 and 1 <= m <= n.

Edited by Johannes W. Meijer, Sep 22 2012

A163936 Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 6, 8, 1, 0, 24, 58, 22, 1, 0, 120, 444, 328, 52, 1, 0, 720, 3708, 4400, 1452, 114, 1, 0, 5040, 33984, 58140, 32120, 5610, 240, 1, 0, 40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0, 362880, 3733920, 11026296, 12440064, 5765500, 1062500

Offset: 1

Johannes W. Meijer, Aug 13 2009

The asymptotic expansions of the higher-order exponential integral E(x,m=1,n) lead to triangle A130524, see A163931 for information on E(x,m,n). The o.g.f.s. of the right-hand columns of triangle A130534 have a nice structure: gf(p) = W1(z,p)/(1-z)^(2*p-1) with p = 1 for the first right-hand column, p = 2 for the second right-hand column, etc. The coefficients of the W1(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. Our triangle is the same as A112007 with an extra right-hand column, see also the second Eulerian triangle A008517. The row sums of our triangle lead to A001147.
We observe that the row sums of the triangles A163936 (m=1), A163937 (m=2), A163938 (m=3) and A163939 (m=4) for z=1 lead to A001147, A001147 (minus a(0)), A001879 and A000457 which are the first four left-hand columns of the triangle of the Bessel coefficients A001497 or, if one wishes, the right-hand columns of A001498. We checked this phenomenon for a few more values of m and found that this pattern persists: m = 5 leads to A001880, m=6 to A001881, m=7 to A038121 and m=8 to A130563 which are the next left- (right-) hand columns of A001497 (A001498). An interesting phenomenon.
If one assumes the triangle not (1,1) based but (0,0) based, one has T(n, k) = E2(n, n-k), where E2(n, k) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 12 2021

			Triangle starts:
[ 1]      1;
[ 2]      1,       0;
[ 3]      2,       1,      0;
[ 4]      6,       8,      1,      0;
[ 5]     24,      58,     22,      1,      0;
[ 6]    120,     444,    328,     52,      1,     0;
[ 7]    720,    3708,   4400,   1452,    114,     1,   0;
[ 8]   5040,   33984,  58140,  32120,   5610,   240,   1,  0;
[ 9]  40320,  341136, 785304, 644020, 195800, 19950, 494,  1, 0;
The first few W1(z,p) polynomials are
W1(z,p=1) = 1/(1-z);
W1(z,p=2) = (1 + 0*z)/(1-z)^3;
W1(z,p=3) = (2 + 1*z + 0*z^2)/(1-z)^5;
W1(z,p=4) = (6 + 8*z + 1*z^2 + 0*z^3)/(1-z)^7.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Row sums equal A001147.
A000142, A002538, A002539, A112008, A112485 are the first few left hand columns.
A000007, A000012, A005803(n+2), A004301, A006260 are the first few right hand columns.
Cf. A163931 (E(x,m,n)), A048994 (Stirling1) and A008517 (Euler).
Cf. A112007, A163937 (E(x,m=2,n)), A163938 (E(x,m=3,n)) and A163939 (E(x,m=4,n)).
Cf. A001497 (Bessel), A001498 (Bessel), A001147 (m=1), A001147 (m=2), A001879 (m=3) and A000457 (m=4), A001880 (m=5), A001881 (m=6) and A038121 (m=7).
Cf. A340556.

with(combinat): a := proc(n, m): add((-1)^(n+k+1)*binomial(2*n-1, k)*stirling1(m+n-k-1, m-k), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..9);  # Johannes W. Meijer, revised Nov 27 2012

Table[Sum[(-1)^(n + k + 1)*Binomial[2*n - 1, k]*StirlingS1[m + n - k - 1, m - k], {k, 0, m - 1}], {n, 1, 10}, {m, 1, n}] // Flatten (* G. C. Greubel, Aug 13 2017 *)

for(n=1,10, for(m=1,n, print1(sum(k=0,m-1,(-1)^(n+k+1)* binomial(2*n-1,k)*stirling(m+n-k-1,m-k, 1)), ", "))) \\ G. C. Greubel, Aug 13 2017

\\ assuming offset = 0:
E2poly(n,x) = if(n == 0, 1, x*(x-1)^(2*n)*deriv((1-x)^(1-2*n)*E2poly(n-1,x)));
{ for(n = 0, 9, print(Vec(E2poly(n,x)))) } \\ Peter Luschny, Feb 12 2021

a(n, m) = Sum_{k=0..(m-1)} (-1)^(n+k+1)*binomial(2*n-1,k)*Stirling1(m+n-k-1,m-k), for 1 <= m <= n.

Assuming offset = 0 the T(n, k) are the coefficients of recursively defined polynomials. T(n, k) = [x^k] x^n*E2poly(n, 1/x), where E2poly(n, x) = x*(x - 1)^(2*n)*d_{x}((1 - x)^(1 - 2*n)*E2poly(n - 1, x))) for n >= 1 and E2poly(0, x) = 1. - Peter Luschny, Feb 12 2021

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

N. J. A. Sloane

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

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
Index entries for sequences related to Hermite polynomials

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

[Factorial(2*n)/(2*Factorial(n-1)): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011

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

Table[(2*n)!/(2*(n-1)!),{n,1,20}] (* Vincenzo Librandi, Nov 22 2011 *)

combinat::catalan(n)*binomial(n+1,2)*n! $ n = 1..16; // Zerinvary Lajos, Feb 15 2007

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

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

A163938 Triangle related to the o.g.f.s. of the right hand columns of A163932 (E(x, m=3, n)).

Original entry on oeis.org

1, 3, 3, 11, 28, 6, 50, 225, 135, 10, 274, 1858, 2092, 486, 15, 1764, 16464, 29148, 13482, 1491, 21, 13068, 158352, 398640, 301220, 70485, 4152, 28, 109584, 1655172, 5552724, 6132780, 2432070, 322971, 10863, 36

Offset: 1

Johannes W. Meijer, Aug 13 2009

The asymptotic expansions of the higher order exponential integral E(x, m=3, n) lead to triangle A163932, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A163932 have a nice structure Gf(p) = W3(z,p)/(1-z)^(2*p+1) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc. The coefficients of the W3(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. The row sums of this triangle lead to A001879, see A163936 for more information.

			The first few W3(z,p) polynomials are:
W3(z,p=1) = 1/(1-z)^3
W3(z,p=2) = (3 + 3*z)/(1-z)^5
W3(z,p=3) = (11 + 28*z + 6*z^2)/(1-z)^7
W3(z,p=4) = (50 + 225*z + 135*z^2 + 10*z^3)/(1-z)^9

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Row sums equal A001879.
A000254 equals the first left hand column.
A000217 equals the first right hand column.
Cf. A163931 (E(x,m,n)) and A163932.
Cf. A163936 (E(x,m=1,n)), A163937 (E(x,m=2,n)) and A163939 (E(x,m=4,n)).

with(combinat): a := proc(n, m): add((-1)^(n+k+1)*((m-k+1)*(m-k)/2!)*binomial(2*n+1, k)*stirling1(m+n-k, m-k+1), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..8); # Johannes W. Meijer, revised Nov 27 2012

Table[Sum[(-1)^(n + k + 1)*Binomial[m - k + 1, 2]*Binomial[2*n + 1, k]*StirlingS1[m + n - k, m - k + 1], {k, 0, m - 1}], {n, 1, 50}, {m, 1, n}] // Flatten (* G. C. Greubel, Aug 13 2017 *)

for(n=1,10, for(m=1,n, print1(sum(k=0,m-1, (-1)^(n+k+1)* binomial(m-k+1,2)*binomial(2*n+1,k) *stirling(m+n-k,m-k+1, 1)) ,", "))) \\ G. C. Greubel, Aug 13 2017

a(n,m) = Sum_{k=0..(m-1)} (-1)^(n+k+1)*binomial(m-k+1,2) *binomial(2*n+1,k) *stirling1(m+n-k,m-k+1), for 1 <= m <= n.

A288950 Number of relaxed compacted binary trees of right height at most one with empty initial and final sequence on level 0.

Original entry on oeis.org

1, 0, 1, 2, 15, 140, 1575, 20790, 315315, 5405400, 103378275, 2182430250, 50414138775, 1264936572900, 34258698849375, 996137551158750, 30951416768146875, 1023460181133390000, 35885072600989486875, 1329858572860198631250, 51938365373373313209375

Offset: 0

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. The number of unbounded relaxed compacted binary trees of size n is A082161(n). The number of relaxed compacted binary trees of right height at most one of size n is A001147(n). See the Genitrini et al. and Wallner link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where node 3 is at depth 1 on the right of node 2 and where the node n+1 has a left sibling. See the Wallner link. - Michael Wallner, Apr 20 2017

			Denote by L the leaf and by o nodes. Every node has exactly two out-going edges or pointers. Internal edges are denoted by - or |. Pointers are omitted and may point to any node further right. The root is at level 0 at the very left.
The general structure is
  L-o-o-o-o-o-o-o-o-o
    |       |     | |
    o   o-o-o   o-o o.
For n=0 the a(0)=1 solution is L.
For n=1 we have a(1)=0 because we need nodes on level 0 and level 1.
For n=2 the a(2)=1 solution is
     L-o
       |
       o
and the pointers of the node on level 1 both point to the leaf.
For n=3 the a(3)=2 solutions have the structure
     L-o
       |
     o-o
where the pointers of the last node have to point to the leaf, but the pointer of the next node has 2 choices: the leaf of the previous node.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A177145, A213527, A288950, A288952, A288953, A288954 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A001879.

terms = 21; (z + (1 - z)/3*(2 - z + (1 - 2z)^(-1/2)) + O[z]^terms // CoefficientList[#, z] &) Range[0, terms-1]! (* Jean-François Alcover, Dec 04 2018 *)

E.g.f.: z + (1-z)/3 * (2-z + (1-2*z)^(-1/2)).
From Seiichi Manyama, Apr 26 2025: (Start)
a(n) = (n-1)*(2*n-3)/(n-2) * a(n-1) for n > 3.
a(n) = A001879(n-2)/3 for n > 2. (End)

A142970 Numerators of n-th approximants of a continued fraction for Pi-3.

Original entry on oeis.org

0, 1, 6, 61, 660, 8901, 133266, 2303865, 43808040, 928665225, 21386693790, 537861526965, 14540730176700, 423407835413325, 13140639311294250, 434929825450371825, 15237733330856005200, 565064979900590948625, 22056613209702152061750, 905913636742121921038125

Offset: 0

Wolfdieter Lang, Sep 15 2008

The corresponding denominators are A001879(n), n >= 0.
Pi = 3 + 1^2/(6 + 3^2/(6 + 5^2/(6 + ... ))). See the J.-P. Delahaye reference. R. Rosenthal mentioned this continued fraction in an e-mail to the author Jul 16 2008.
For the approximants in lowest terms cf. the ones for 3*(Pi-3) given by A130411(n)/A130412(n) in lowest terms.
The above continued fraction for Pi is the particular case n = 0, x = 3 of a result of Ramanujan, previously given by Euler - see Berndt et al., Chapter 12, Entry 25, p. 268. - Peter Bala, Feb 19 2015

			Approximants a(n)/A001879(n) (not in lowest terms): [0/1]; [1/6]; [6/45]; [61/420]; [660/4725]; [8901/62370];..
Approximants in lowest terms: [0/1]; [1/6]; [2/15]; [61/420]; [44/315]; [989/6930]; ...

J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997. In German: Pi - die Story, Birkhäuser, 1999 Basel, p. 87.

B. C. Berndt, R. L. Lamphere, and B. M. Wilson Chapter 12 of Ramanujan's second notebook: Continued fractions, Rocky Mountain Journal of Mathematics, Volume 15, Number 2 (1985), 235-310
L. Euler, De fractionibus continuis observationes, The Euler Archive, Index Number 123, Section 67.
Wolfdieter Lang, Approximants for Pi-3 and more
L. J. Lange, An Elegant Continued Fraction for π, The American Mathematical Monthly, 106 (1999), 456-458.

Cf. A000796, A001879, A130411, A130412, A254795.

I:=[1,6]; [0] cat [n le 2 select I[n] else 6*Self(n-1)+(2*n-1)^2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 20 2015

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==6 a[n-1] + (2 n-1)^2 a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Feb 20 2015 *)

a(n) = 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(0)=0, a(1)=1.
E.g.f.: (-3*(1+x-sqrt(1-4*x^2))+ 2*(1+x)*arcsin(2*x))/(1-2*x)^(5/2) from the solution of the linear second order differential equation (1-4*x^2)*y''(x) - 2*(8*x+3)*y'(x) - 9*y(x)=0, obtained from the recurrence, with inputs y(0)=0 and y'(0)=1. A special solution is the e.g.f. of the denominators A001879: (1+x)/(1-2*x)^(5/2).
a(n) ~ (Pi-3) * 2^(n+3/2) * n^(n+2) / exp(n). - Vaclav Kotesovec, Oct 05 2013

A288952 Number of relaxed compacted binary trees of right height at most one with empty sequences between branch nodes on level 0.

Original entry on oeis.org

1, 0, 1, 2, 15, 92, 835, 8322, 99169, 1325960, 19966329, 332259290, 6070777999, 120694673748, 2594992240555, 59986047422378, 1483663965460545, 39095051587497488, 1093394763005554801, 32347902448449172530, 1009325655965539561231, 33125674098690460236620

Offset: 0

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels a maximal young leaf has to be followed by a non-maximal young leaf. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

			See A288950 and A288953.

Muniru A Asiru, Table of n, a(n) for n = 0..100
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A177145, A213527, A288950, A288953, A288954 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).

a := [1,0];; for n in [3..10^2] do a[n] := (n-2)*a[n-1] + (n-2)^2*a[n-2]; od; a; # Muniru A Asiru, Jan 26 2018

a:=proc(n) option remember: if n=0 then 1 elif n=1 then 0 elif n>=2 then (n-1)*procname(n-1)-(n-1)^2*procname(n-2) fi; end:
seq(a(n),n=0..100); # Muniru A Asiru, Jan 26 2018

Fold[Append[#1, (#2 - 1) Last[#1] + #1[[#2 - 1]] (#2 - 1)^2] &, {1, 0}, Range[2, 21]] (* Michael De Vlieger, Jan 28 2018 *)

E.g.f.: exp( -Sum_{n>=1} Fibonacci(n-1)*x^n/n ), where Fibonacci(n) = A000045(n).
E.g.f.: exp( -1/sqrt(5)*arctanh(sqrt(5)*z/(2-z)) )/sqrt(1-z-z^2).
a(0) = 1, a(1) = 0, a(n) = (n-1)*a(n-1) + (n-1)^2*a(n-2). - Daniel Suteu, Jan 25 2018

cp's OEIS Frontend

A001498 Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).

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A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

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A163936 Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).

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

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A163938 Triangle related to the o.g.f.s. of the right hand columns of A163932 (E(x, m=3, n)).

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A288950 Number of relaxed compacted binary trees of right height at most one with empty initial and final sequence on level 0.

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A038121 E.g.f.: (1 + 15x + (45/2)x^2 + (5/2)x^3)/(1 - 2x)^(13/2).