A000906 -id:A000906 - cp's OEIS frontend

A000457 Exponential generating function: (1+3x)/(1-2x)^(7/2).

1, 10, 105, 1260, 17325, 270270, 4729725, 91891800, 1964187225, 45831035250, 1159525191825, 31623414322500, 924984868933125, 28887988983603750, 959493919812553125, 33774185977401870000, 1255977541034632040625

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 10*x + 105*x^2 + 1260*x^3 + 17325*x^4 + 270270*x^5 + ... - _Michael Somos_, Dec 15 2023

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
C. Jordan, Calculus of Finite Differences. Eggenberger, Budapest and Röttig-Romwalter, Sopron 1939; Chelsea, NY, 1965, p. 172.
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).

G. C. Greubel, Table of n, a(n) for n = 0..200
Selden Crary, Richard Diehl Martinez, and Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 1.
H. W. Gould, Harris Kwong, and Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), #15.9.6.
C. Jordan, On Stirling's Numbers, Tohoku Math. J., 37 (1933), 254-278.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
J. Riordan, Notes to N. J. A. Sloane, Jul. 1968
Eric Weisstein's World of Mathematics, Stirling Number of the First Kind.

Equals (1/2)*A000906.
Third column of triangle A001497.
Second column (m=1) of unsigned Laguerre-Sonin a=1/2 triangle |A130757|.
Diagonal k=n-1 of triangle A134991.
Cf. A160473, A163939.

[Factorial(2*n+3)/(6*Factorial(n)*2^n): n in [0..30]]; // G. C. Greubel, May 15 2018

Table[(2n+3)!/(3!*n!*2^n), {n,0,30}] (* G. C. Greubel, May 15 2018 *)

for(n=0, 30, print1((2*n+3)!/(3!*n!*2^n), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = (2n+3)!/( 3!*n!*2^n ).
a(n) = (n+1)*(2*n+3)!!/3, n>=0, with (2*n+3)!! = A001147(n+2).
a(n) = Sum_{j=0..n} (j + 1) * Eulerian2(n + 2, n - j). - Peter Luschny, Feb 13 2023

More terms from Sascha Kurz, Aug 15 2002

A111999 T(n, k) = [x^k] (-1)^nSum_{k=0..n} E2(n, n-k)(1+x)^(n-k) where E2(n, k) are the second-order Eulerian numbers. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= n.

Original entry on oeis.org

-1, 3, 2, -15, -20, -6, 105, 210, 130, 24, -945, -2520, -2380, -924, -120, 10395, 34650, 44100, 26432, 7308, 720, -135135, -540540, -866250, -705320, -303660, -64224, -5040, 2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320, -34459425, -183783600, -416215800

Offset: 1

Wolfdieter Lang, Sep 12 2005

Previous name was: A triangle that converts certain binomials into triangle A008276 (diagonals of signed Stirling1 triangle A008275).
Stirling1(n,n-m) = A008275(n,n-m) = Sum_{k=0..m-1}a(m,k)*binomial(n,2*m-k).
The unsigned column sequences start with A001147, A000906 = 2*A000457, 2*|A112000|, 4*|A112001|.
The general results on the convolution of the refined partition polynomials of A133932, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these unsigned polynomials. - Tom Copeland, Sep 20 2016

			Triangle starts:
  [1]      -1;
  [2]       3,       2;
  [3]     -15,     -20,       -6;
  [4]     105,     210,      130,       24;
  [5]    -945,   -2520,    -2380,     -924,     -120;
  [6]   10395,   34650,    44100,    26432,     7308,     720;
  [7] -135135, -540540,  -866250,  -705320,  -303660,  -64224,  -5040;
  [8] 2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320.

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 152. Table C_{m, nu}.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, page 4.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
EqWorld, Integral Transforms
David J. Jeffrey, G. A. Kalugin, N. Murdoch, Lagrange inversion and Lambert W, 17th Int'l Symp. Symb. Numer. Algor. Sci. Comp. (SYNASC 2015).
Wolfdieter Lang, First 10 rows.
Lajos Takács, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581. Table 3 gives an unsigned version of the triangle.

Row sums give A032188(m+1)*(-1)^m, m>=1. Unsigned row sums give A032188(m+1), m>=1.
Cf. A008517 (second-order Eulerian triangle) for a similar formula for |Stirling1(n, n-m)|.
Cf. A000457, A000906, A001147, A008275, A008276, A008306, A112001, A133932,

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
E2 := (n, k) -> combinat[eulerian2](n, k):
poly := n -> (-1)^n*add(E2(n, n-k)*(1+x)^(n-k), k = 0..n):
seq(CoeffList(poly(n)), n = 1..8); # Peter Luschny, Feb 05 2021

a[m_, k_] := a[m, k] = Which[m < k + 1, 0, And[m == 1, k == 0], -1, k == -1, 0, True, -(2 m - k - 1)*(a[m - 1, k] + a[m - 1, k - 1])]; Table[a[m, k], {m, 9}, {k, 0, m - 1}] // Flatten (* Michael De Vlieger, Sep 23 2016 *)

a(m, k)=0 if m

From Tom Copeland, May 05 2010 (updated Sep 12 2011): (Start)

The integral from 0 to infinity w.r.t. w of

exp[-w(u+1)] (1+u*z*w)^(1/z) gives a power series, f(u,z), in z for reversed row polynomials in u of A111999, related to an Euler transform of diagonals of A008275.

Let g(u,x) be obtained from f(u,z) by replacing z^n with x^(n+1)/(n+1)!;

g(u,x)= x - u^2 x^2/2! + (2 u^3 + 3 u^4) x^3/3! - (6 u^4 + 20 u^5 + 15 u^6) x^4/4! + ... , an e.g.f. associated to f(u,z).

Then g^(-1)(u,x)=(1+u)*x - log(1+u*x) is the comp. inverse of g(u,x) in x, and, consequently, A133932 is a refinement of A111999.

With h(u,x)= 1/(dg^(-1)/dx)= (1+u*x)/(1+(1+u)*u*x),

g(u,x)=exp[x*h(u,t)d/dt] t, evaluated at t=0. Also, dg(u,x)/dx = h(u,g(u,x)). (End)

From Tom Copeland, May 06 2010: (Start)

For m,k>0, a(m,k) = Sum(j=2 to 2m-k+1): (-1)^(2m-k+1+j) C(2m-k+1,j) St1d(j,m),

where C(n,j) is the binomial coefficient and St1d(j,m) is the (j-m)-th element of the m-th subdiagonal of A008275 for (j-m)>0 and is 0 otherwise,

e.g., St1d(1,1) = 0, St1d(2,1) = -1, St1d(3,1) = -3, St1d(4,1) = -6. (End)

From Tom Copeland, Sep 03 2011 (updated Sep 12 2011): (Start)

The integral from 0 to infinity w.r.t. w of

exp[-w*(u+1)/u] (1+u*z*w)^(1/(u^2*z)) gives a power series, F(u,z), in z for the row polynomials in u of A111999.

Let G(u,x) be obtained from F(u,z) by replacing z^n with x^(n+1)/(n+1)!;

G(u,x) = x - x^2/2! + (3 + 2 u) x^3/3! - (15 + 20 u + 6 u^2) x^4/4! + ... , an e.g.f. for A111999 associated to F(u,z).

G^(-1)(u,x) = ((1+u)*u*x - log(1+u*x))/u^2 is the comp. inverse of G(u,x) in x.

With H(u,x) = 1/(dG^(-1)/dx) = (1+u*x)/(1+(1+u)*x),

G(u,x) = exp[x*H(u,t)d/dt] t, evaluated at t=0. Also, dG(u,x)/dx = H(u,G(u,x)). (End)

From Tom Copeland, Sep 16 2011: (Start)

f(u,z) and F(u,z) are expressible in terms of the incomplete gamma function Γ(v,p)(see Laplace Transforms for Power-law Functions at EqWorld):

With K(p,s) = p^(-s-1) exp(p) Γ(s+1,p),

f(u,z) = K(p,s)/(u*z) with p=(u+1)/(u*z) and s=1/z , and

F(u,z) = K(p,s)/(u*z) with p=(u+1)/(u^2*z) and s=1/(u^2*z). (End)

Diagonals of A008306 are reversed rows of A111999 (see P. Bala). - Tom Copeland, May 08 2012

New name from Peter Luschny, Feb 05 2021

A098503 Triangle T(n,k) by rows: coefficient [x^(n-k)] of 2^n * n! *L(n,1/2,x), with L the generalized Laguerre polynomials in the Abramowitz-Stegun normalization.

Original entry on oeis.org

1, -2, 3, 4, -20, 15, -8, 84, -210, 105, 16, -288, 1512, -2520, 945, -32, 880, -7920, 27720, -34650, 10395, 64, -2496, 34320, -205920, 540540, -540540, 135135, -128, 6720, -131040, 1201200, -5405400, 11351340, -9459450, 2027025, 256, -17408

Offset: 0

Ralf Stephan, Sep 15 2004

			2^0 *0! *L(0,1/2,x) = 1.
2^1 *1! *L(1,1/2,x) = -2*x + 3.
2^2 *2! *L(2,1/2,x) = 4*x^2 - 20*x + 15.
2^3 *3! *L(3,1/2,x) = -8*x^3 + 84*x^2 - 210*x + 105.
2^4 *4! *L(4,1/2,x) = 16*x^4 - 288*x^3 + 1512*x^2 - 2520*x + 945.
Triangle begins:
    1;
   -2,     3;
    4,   -20,    15;
   -8,    84,  -210,     105;
   16,  -288,  1512,   -2520,    945;
  -32,   880, -7920,   27720, -34650,   10395;
   64, -2496, 34320, -205920, 540540, -540540, 135135;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
R. J. Mathar, Gauss-Laguerre and Gauss-Hermite quadrature on 64, 96 and 128 nodes, viXra:1303.0013, Section 3.

Columns include (-1)^n times A000079, n/2*A014480. Diagonals include A001147, -A000906, 4*A001881.

Cf. A223168..A223172 and A223523..A223532.

Table[Reverse[Table[2^n*(-1)^k*n!/k!*Binomial[n + 1/2, n - k], {k, 0, n}]], {n, 0, 7}] (* T. D. Noe, Apr 05 2013 *)

T(n, k) = (-2)^n * (-1)^k * n!/(n-k)! * binomial(n+1/2,k), = (-1)^(n+k) *2^(n-2k) *k! *binomial(2n+1,2k)*binomial(2k,k), n>=0, k<=n.

A162974 Triangle read by rows: T(n,k) is the number of derangements of {1,2,...,n} having k cycles of length 2 (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 6, 0, 3, 24, 20, 0, 160, 90, 0, 15, 1140, 504, 210, 0, 8988, 4480, 1260, 0, 105, 80864, 41040, 9072, 2520, 0, 809856, 404460, 100800, 18900, 0, 945, 8907480, 4447520, 1128600, 166320, 34650, 0, 106877320, 53450496, 13347180, 2217600

Offset: 0

Emeric Deutsch, Jul 22 2009

Row n has 1 + floor(n/2) entries.
Sum of entries in row n = A000166(n) (the derangement numbers).
T(n,0) = A038205(n).
Sum_{k>=0} k*T(n,k) = A000387(n).

			T(4,2)=3 because we have (12)(34), (13)(24), and (14)(23).
Triangle starts:
    1;
    0;
    0,  1;
    2,  0;
    6,  0,  3;
   24, 20,  0;
  160, 90,  0, 15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000166, A000387, A038205, A114320.
T(2n,n) gives A001147.
T(2n+3,n) gives A000906(n) = 2*A000457(n).

G := exp((1/2)*z*(t*z-z-2))/(1-z): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 13 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*n)) end do;
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add((j-1)!*
      `if`(j=2, x, 1)*b(n-j)*binomial(n-1, j-1), j=2..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..14);  # Alois P. Heinz, Jan 27 2022

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[(j - 1)!*If[j == 2, x, 1]*b[n - j]*Binomial[n - 1, j - 1], {j, 2, n}]]];
T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Sep 17 2024, after Alois P. Heinz *)

E.g.f.: G(t,z) = exp(z(tz-z-2)/2)/(1-z).

A112000 One half of third column (k=2) of triangle A111999.

Original entry on oeis.org

-3, 65, -1190, 22050, -433125, 9144135, -208107900, 5099994900, -134219460375, 3781060408125, -113633468798850, 3631422078033750, -123022987568105625, 4405418319999571875, -166312279434175875000, 6602853358582065585000, -275059081486584416896875

Offset: 0

Wolfdieter Lang, Sep 12 2005

Cf. Second (k=1) column: A000906(n+2)*(-1)^n = 2*A000457(n+2)*(-1)^n, n>=0.

a(n)=A111999(n+3, 2)/2, n>=0.

Conjecture: +n*(4*n+5)*a(n) +(2*n+3)*(n+2)*(4*n+9)*a(n-1)=0. - R. J. Mathar, Jul 09 2017

Showing 1-5 of 5 results.

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A000457 Exponential generating function: (1+3*x)/(1-2*x)^(7/2).

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A111999 T(n, k) = [x^k] (-1)^n*Sum_{k=0..n} E2(n, n-k)*(1+x)^(n-k) where E2(n, k) are the second-order Eulerian numbers. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= n.

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A098503 Triangle T(n,k) by rows: coefficient [x^(n-k)] of 2^n * n! *L(n,1/2,x), with L the generalized Laguerre polynomials in the Abramowitz-Stegun normalization.

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A162974 Triangle read by rows: T(n,k) is the number of derangements of {1,2,...,n} having k cycles of length 2 (0 <= k <= floor(n/2)).

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A112000 One half of third column (k=2) of triangle A111999.

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A000457 Exponential generating function: (1+3x)/(1-2x)^(7/2).

A111999 T(n, k) = [x^k] (-1)^nSum_{k=0..n} E2(n, n-k)(1+x)^(n-k) where E2(n, k) are the second-order Eulerian numbers. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= n.