A008275 -id:A008275 - cp's OEIS frontend

A000914 Stirling numbers of the first kind: s(n+2, n).

0, 2, 11, 35, 85, 175, 322, 546, 870, 1320, 1925, 2717, 3731, 5005, 6580, 8500, 10812, 13566, 16815, 20615, 25025, 30107, 35926, 42550, 50050, 58500, 67977, 78561, 90335, 103385, 117800, 133672, 151096, 170170, 190995, 213675, 238317, 265031

Offset: 0

N. J. A. Sloane

Sum of product of unordered pairs of numbers from {1..n+1}.
Number of edges of a complete k-partite graph of order k*(k+1)/2 (A000217), K_1,2,3,...,k. - Roberto E. Martinez II, Oct 18 2001
This sequence holds the x^(n-2) coefficient of the characteristic polynomial of the N X N matrix A formed by MAX(i,j), where i is the row index and j is the column index of element A[i][j], 1 <= i,j <= N. Here N >= 2. - Paul Max Payton, Sep 06 2005
The sequence contains the partial sums of A006002, which represent the areas beneath lines created by the triangular numbers plotted (t(1),t(2)) connected to (t(2),t(3)) then (t(3),t(4))...(t(n-1),t(n)) and the x-axis. - J. M. Bergot, May 05 2012
Number of functions f from [n+2] to [n+2] with f(x)=x for exactly n elements x of [n+2] and f(x)>x for exactly two elements x of [n+2]. To prove this, let the two elements of [n+2] with a larger image be labeled i and j. Note both i and j must be less than n+2. Then there are (n+2-i) choices for f(i) and (n+2-j) choices for f(j). Summing the product of the number of choices over all sets {i,j} gives us "Sum of product of unordered pairs of numbers from {1..n+1}" in the first line of the Comments Section. See the example in the Example Section below. - Dennis P. Walsh, Sep 06 2017
Zhu Shijie gives in his Magnus Opus "Jade Mirror of the Four Unknowns" the problem: "Apples are piled in the form of a triangular pyramid. The top apple is worth 2 and the price of the whole is 1320. Each apple in one layer costs 1 less than an apple in the next layer below." We find the solution 9 to this problem in this sequence 1320 = a(9). Zhu Shijie gave the solution polynomial: "Let the element tian be the number of apples in a side of the base. From the statement we have 31680 for the negative shi, 10 for the positive fang, 21 for the positive first lian, 14 for the positive last lian, and 3 for the positive yu."  This translates into the polynomial equation: 3*x^4 + 14*x^3 + 21*x^2 + 10*x - 31680 = 0. - Thomas Scheuerle, Feb 10 2025

			Examples include E(K_1,2,3) = s(2+2,2) = 11 and E(K_1,2,3,4,5) = s(4+2,4) = 85, where E is the function that counts edges of graphs.
For n=2 the a(2)=11 functions f:[4]->[4] with exactly two f(x)=x and two f(x)>x are given by the 11 image vectors of form <f(1),f(2),f(3),f(4)> that follow: <1,3,4,4>, <1,4,4,4>, <2,2,4,4>, <3,2,4,4>, <4,2,4,4>, <2,3,3,4>, <2,4,3,4>, <3,3,3,4>, <3,4,3,4>, <4,3,3,4>, and <4,4,3,4>. - _Dennis P. Walsh_, Sep 06 2017

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
George E. Andrews, Number Theory, Dover Publications, New York, 1971, p. 4.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
H. S. Hall and S. R. Knight, Higher Algebra, Fourth Edition, Macmillan, 1891, p. 518.
Zhu Shijie, Jade Mirror of the Four Unknowns (Siyuan yujian), Book III Guo Duo Die Gang (Piles of Fruit), Problem number 1, 1303.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
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].
Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., Vol. 1, No. 3 (1926), pp. 44-49. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Zhu Shijie, Jade Mirror of the Four Unknowns 2, Translation by Library of Chinese classics, original from 1303.
Wikipedia, Jade Mirror of the Four Unknowns.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000290, A033428, A033581, A033583, A008275, A052149.
Cf. similar sequences listed in A241765.
Cf. A001296.
Cf. A006325(n+1) (Zhu Shijie's problem number 2 uses a pyramid with square base).

a000914 n = a000914_list !! n
a000914_list = scanl1 (+) a006002_list
-- Reinhard Zumkeller, Mar 25 2014

[StirlingFirst(n+2, n): n in [0..40]]; // Vincenzo Librandi, May 28 2019

A000914 := n -> 1/24*(n+1)*n*(n+2)*(3*n+5);
A000914 := proc(n)
    combinat[stirling1](n+2,n) ;
end proc: # R. J. Mathar, May 19 2016

Table[StirlingS1[n+2,n],{n,0,40}] (* Harvey P. Dale, Aug 24 2011 *)
a[ n_] := n (n + 1) (n + 2) (3 n + 5) / 24; (* Michael Somos, Sep 04 2017 *)

a(n)=sum(i=1,n+1,sum(j=1,n+1,i*j*(i

a(n)=sum(i=1,n+1,sum(j=1,i-1,i*j)) \\ Charles R Greathouse IV, Apr 07 2015

a(n) = binomial(n+2, 3)*(3*n+5)/4 \\ Charles R Greathouse IV, Apr 07 2015

[stirling_number1(n+2, n) for n in range(41)] # Zerinvary Lajos, Mar 14 2009

a(n) = binomial(n+2, 3)*(3*n+5)/4 = (n+1)*n*(n+2)*(3*n+5)/24.
E.g.f.: exp(x)*x*(48 + 84*x + 32*x^2 + 3*x^3)/24.
G.f.: (2*x+x^2)/(1-x)^5. - Simon Plouffe in his 1992 dissertation.
a(n) = Sum_{i=1..n} i*(i+1)^2/2. - Jon Perry, Jul 31 2003
a(n) = A052149(n+1)/2. - J. M. Bergot, Jun 02 2012
-(3*n+2)*(n-1)*a(n) + (n+2)*(3*n+5)*a(n-1) = 0. - R. J. Mathar, Apr 30 2015
a(n) = a(n-1) + (n+1)*binomial(n+1,2) for n >= 1. - Dennis P. Walsh, Sep 21 2015
a(n) = A001296(-2-n) for all n in Z. - Michael Somos, Sep 04 2017
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 162*log(3)/5 - 18*sqrt(3)*Pi/5 - 384/25.
Sum_{n>=1} (-1)^(n+1)/a(n) = 36*sqrt(3)*Pi/5 - 96*log(2)/5 - 636/25. (End)
a(n) = 3*A000332(n+3) - A000292(n). - Yasser Arath Chavez Reyes, Apr 03 2024

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
Comments from Michael Somos, Jan 29 2000
Erroneous duplicate of the polynomial formula removed by R. J. Mathar, Sep 15 2009

A132440 Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0

Offset: 0

Tom Copeland, Nov 13 2007, Nov 15 2007, Nov 22 2007, Dec 02 2007

Let M(t) = exp(t*T) = lim_{n->oo} (1 + t*T/n)^n.
Pascal matrix = [ binomial(n,k) ] = M(1) = exp(T), truncating the series gives the n X n submatrices.
Inverse Pascal matrix = M(-1) = exp(-T) = matrix for inverse binomial transform.
A(j) = T^j / j! equals the matrix [binomial(n,k) * delta(n-k-j)] where delta(n) = 1 if n=0 and vanishes otherwise (Kronecker delta); i.e., A(j) is a matrix with all the terms 0 except for the j-th lower (or main for j=0) diagonal, which equals that of the Pascal triangle. Hence the A(j)'s form a linearly independent basis for all matrices of the form [binomial(n,k) * d(n-k)] which include as a subset the invertible associated matrices of the list partition transform (LPT) of A133314.
For sequences with b(0) = 1, umbrally,
M[b(.)] = exp(b(.)*T) = [ binomial(n,k) * b(n-k) ] = matrices associated to b by LPT.
[M[b(.)]]^(-1) = exp(c(.)*T) = [ binomial(n,k) * c(n-k) ] = matrices associated to c, where c = LPT(b) . Or,
[M[b(.)]]^(-1) = exp[LPT(b(.))*T] = LPT[M(b(.))] = M[LPT(b(.))]= M[c(.)].
This is related to xDx, the iterated Laguerre transform and the general Euler transformation of a sequence through the comments in A132013 and A132014 and the relation [Sum_{k=0..n} binomial(n,k) * b(n-k) * d(k)] = M(b)*d, (n-th term). See also A132382.
If b(n,x) is a binomial type Sheffer sequence, then M[b(.,x)]*s(y) = s(x+y) when s(y) = (s(0,y),s(1,y),s(2,y),...) is an array for a Sheffer sequence with the same delta operator as b(n,x) and [M[b(.,x)]]^(-1) is given by the formulas above with b(n) replaced by b(n,x) as b(0,x)=1 for a binomial-type Sheffer sequence.
T = I - A132013 and conversely A132013 = I - T, which is the matrix representation for the iterated mixed order Laguerre transform characterized in A132013 (and A132014).
(I-T)^m generates the group [A132013]^m for m = 0,1,2,... discussed in A132014.
The inverse is 1/(I-T) = I + T + T^2 + T^3 + ... = [A132013]^(-1) = A094587 with the associated sequence (0!,1!,2!,3!,...) under the LPT.
And 1/(I-T)^2 = I + 2*T + 3*T^2 + 4*T^3 + ... = [A132013]^(-2) = A132159 with the associated sequence (1!,2!,3!,4!,...) under the LPT.
The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.'s A(x) and B(x), or e.g.f.'s EA(x) and EB(x).
1) b(0) = 0, b(n) = n * a(n-1),
2) B(x) = xDx A(x)
3) B(x) = x * Lag(1,-:xD:) A(x)
4) EB(x) = x * EA(x) where D is the derivative w.r.t. x, (:xD:)^j = x^j*D^j and Lag(n,x) is the Laguerre polynomial.
So the exponentiated operator can be characterized as
5) exp(t*T) A(x) = exp(t*xDx) A(x) = [Sum_{n=0,1,...} (t*x)^n * Lag(n,-:xD:)] A(x) = [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u) ] A(x) (eval. at u=x) = A[x/(1-t*x)]/(1-t*x), a generalized Euler transformation for an o.g.f.,
6) exp(t*T) EA(x) = exp(t*x)*EA(x) = exp[(t+a(.))*x], gen. Euler trf. for an e.g.f.
7) exp(t*T) * a = M(t) * a = [Sum_{k=0..n} binomial(n,k) * t^(n-k) * a(k)].
The umbral extension of formulas 5, 6 and 7 gives formally
8) exp[c(.)*T] A(x) = exp(c(.)*xDx) A(x) = [Sum_{n>=0} (c(.)*x)^n * Lag(n,-:xD:)] A(x) = [exp{[c(.)*u/(1-c(.)*u)]*:xD:} / (1-c(.)*u) ] A(x) (eval. at u=x) = A[x/(1-c(.)*x)]/(1-c(.)*x), where the umbral evaluation should be applied only after a power series in c is obtained,
9) exp[c(.)*T] EA(x) = exp(c(.)*x)*EA(x) = exp[(c(.)+a(.))*x]
10) exp[c(.)*T] * a = M[c(.)] * a = [Sum_{k=0..n} binomial(n,k) * c(n-k) * a(k)] .
The n X n principal submatrix of T is nilpotent, in particular, [Tsub_n]^(n+1) = 0, n=0,1,2,3,....
Note (xDx)^n = x^n D^n x^n = x^n n! (:Dx:)^n/n! = x^n n! Lag(n,-:xD:).
The operator xDx is an important, classical operator explored by among others Dattoli, Al-Salam, Carlitz and Stokes and even earlier investigators.
For a recent treatment of xDx, DxD and more general operators see the paper "Laguerre-type derivatives: Dobinski relations and combinatorial identities". - Karol A. Penson, Sep 15 2009
See Copeland's link for generalized Laguerre functions and connection to fractional differ-integrals in exercises through (:Dx:)^a/a!=(D^a x^a)/a!. - Tom Copeland, Nov 17 2011
From Tom Copeland, Apr 25 2014: (Start)
Conjugation or "similarity" transformations of [dP]=A132440 have an operator interpretation (cf. A074909 and A238363):
In general, select two operators A and B such that A^n = F1(n,B) and B^n = F2(n,A); then A^n = F1(n,F2(.,A)) and B^n = F2(n,F1(.,B)), evaluated umbrally, i.e., F1(n,F2(.,x))=F2(n,F1(.,x))=x^n, implying the polynomials F1 and F2 are an umbral compositional inverse pair.
One such pair are the Bell polynomials Bell(n,x) and falling factorials (x)_n with Bell(n,:xD:)=(xD)^n and (xD)_n=:xD:^n (cf. A074909). Another are the Laguerre polynomials LN(n,x)= n!*Lag(n,x) (A021009), which are umbrally self-inverse, with LN(n,-:xD:)=:Dx:^n and LN(n,:Dx:)= (-:xD:)^n with :Dx:^n=D^n*x^n.
Evaluating, for n>=0, the operator derivative d(B^n)/dA = d(F2(n,A))/dA in the basis B^n, i.e., with A^n finally replaced by F1(n,B), or A^n=F1(.,B)^n=F1(n,B), is equivalent to the matrix conjugation
A)   [F2]*[dP]*[F1]
B) = [F2]*[dP]*[F2]^(-1)
C) = [F1]^(-1)*[dP]*[F1],
where [F1] is the lower triangular matrix with the n-th row the coefficients of F1(n,x) and analogously for [F2].
So, given the row vector Rv=(c0 c1 c2 c3 ...) and the column vector Cv(x)=(1 x x^2 x^3 ...)^Transpose, form the power series V(x)=Rv*Cv(x).
D) dV(B)/dA = Rv * [F2]*[dP]*[F1] * Cv(B).
E) With A=D and B=D, F1(n,x)=F2(n,x)=x^n and [F1]=[F2]=I. Then d(B^n)/dA = d(D^n)/dD = n * D^(n-1); therefore, consistently [F2]*[dP]*[F1] = [dP] and dV(D)/dD = Rv * [dP] * Cv(D). (End)

			Matrix T begins
  0;
  1,0;
  0,2,0;
  0,0,3,0;
  0,0,0,4,0;
  ...

T. Mansour and M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, Chapman and Hall/CRC, 2015, (x^n D^n x^n on p. 187).

Robert Israel, Table of n, a(n) for n = 0..10000
W. A. Al-Salam, Operational representations for the Laguerre and other polynomials, Duke Math. Jour., vol. 31 (1964), pp. 127-142.
Tom Copeland, Fractional Calculus, Gamma Classes, the Riemann Zeta Function, and an Appell Pair of Sequences.
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator.
Tom Copeland, Infinigens, the Pascal Pyramid, and the Witt and Virasoro Algebras.
Tom Copeland, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions (pdf).
Tom Copeland, Mathemagical Forests.
Tom Copeland, Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops.
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009. (Cf. Viennot's Laguerre histoires.)
K. A. Penson, P. Blasiak, A. Horzela, G.H.E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, arXiv:0904.0369 [math-ph], 2009.
K. A. Penson, P. Blasiak, A. Horzela, G.H.E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, Journal of Mathematical Physics vol. 50, (2009) 083512.
G. Stokes, Note on certain formulae in the calculus of operations, Proceedings of the Royal Society of Edinburgh, IX, pp. 101-102, 1876.

seq(op([0$i,i]),i=1..20); # Robert Israel, Oct 02 2015

Table[PadLeft[{n, 0}, n+1], {n, 0, 11}] // Flatten (* Jean-François Alcover, Apr 30 2014 *)

T = log(P) with the Pascal matrix P:=A007318. This should be read as T_N = log(P_N) with P_N the N X N matrix P, N>=2. Because P_N is lower triangular with all diagonal elements 1, the series log(1_N-(1_N-P_N)) stops after N-1 terms because (1_N-P_N)^N is the 0_N-matrix. - Wolfdieter Lang, Oct 14 2010
Given a polynomial sequence p_n(x) with p_0(x)=1 and the lowering and raising operators L and R defined by L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), the matrix T represents the action of R*L*R in the p_n(x) basis. For p_n(x) = x^n, L = D = d/dx and R = x. For p_n(x) = x^n/n!, L = DxD and R = D^(-1). - Tom Copeland, Oct 25 2012
From Tom Copeland, Apr 26 2014: (Start)
A) T = exp(A238385-I) - I
B)   = [St1]*P*[St2] - I
C)   = [St1]*P*[St1]^(-1) - I
D)   = [St2]^(-1)*P*[St2] - I
E)   = [St2]^(-1)*P*[St1]^(-1) - I
where P=A007318, [St1]=padded A008275 just as [St2]=A048993=padded A008277, and I=identity matrix. (End)
From Robert Israel, Oct 02 2015: (Start)
G.f. Sum_{k >= 1} k x^((k+3/2)^2/2 - 17/8) is related to Jacobi theta functions.
If 8*n+17 = y^2 is a square, then a(n) = (y-3)/2, otherwise a(n) = 0. (End)

Missing zero added in table by Tom Copeland, Feb 25 2014

A008276 Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -6, 11, -6, 1, -10, 35, -50, 24, 1, -15, 85, -225, 274, -120, 1, -21, 175, -735, 1624, -1764, 720, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 1, -36, 546, -4536, 22449, -67284, 118124, -109584, 40320, 1, -45

Offset: 1

N. J. A. Sloane

n-th row of the triangle = charpoly of an (n-1) X (n-1) matrix with (1,2,3,...) in the diagonal and the rest zeros. - Gary W. Adamson, Mar 19 2009
From Daniel Forgues, Jan 16 2016: (Start)
For n >= 1, the row sums [of either signed or absolute values] are
  Sum_{k=1..n} T(n,k) = 0^(n-1),
  Sum_{k=1..n} |T(n,k)| = T(n+1,1) = n!. (End)
The moment generating function of the probability density function p(x, m=q, n=1, mu=q) = q^q*x^(q-1)*E(x, q, 1)/(q-1)!, with q >= 1, is M(a, m=q, n=1, mu=q) = Sum_{k=0..q}(A000312(q) / A000142(q-1)) * A008276(q, k) * polylog(k, a) / a^q , see A163931 and A274181. - Johannes W. Meijer, Jun 17 2016
Triangle of coefficients of the polynomial x(x-1)(x-2)...(x-n+1), also denoted as falling factorial (x)n, expanded into decreasing powers of x. - _Ralf Stephan, Dec 11 2016

			3!*binomial(x,3) = x*(x-1)*(x-2) = x^3 - 3*x^2 + 2*x.
Triangle begins
  1;
  1,  -1;
  1,  -3,   2;
  1,  -6,  11,   -6;
  1, -10,  35,  -50,  24;
  1, -15,  85, -225, 274, -120;
...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), p. 257.

T. D. Noe, Rows n=0..100 of triangle, flattened
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].
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See pp. 10, 12.
Bill Gosper, Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7
Eric Weisstein's World of Mathematics, Stirling Number of the First Kind
Wikipedia, Stirling numbers and exponential generating functions

See A008275 and A048994, which are the main entries for this triangle of numbers.
See A008277 triangle of Stirling numbers of the second kind, S2(n,k).
Cf. A054654, A054655, A084938, A145324, A094216, A003422, A000166, A000110, A000204, A000045, A000108.

a008276 n k = a008276_tabl !! (n-1) !! (k-1)
a008276_row n = a008276_tabl !! (n-1)
a008276_tabl = map init $ tail a054654_tabl
-- Reinhard Zumkeller, Mar 18 2014

seq(seq(coeff(expand(n!*binomial(x,n)),x,j),j=n..1,-1),n=1..15); # Robert Israel, Jan 24 2016
A008276 := proc(n, k): combinat[stirling1](n, n-k+1) end: seq(seq(A008276(n, k), k=1..n), n=1..9); # Johannes W. Meijer, Jun 17 2016

len = 47; m = Ceiling[Sqrt[2*len]]; t[n_, k_] = StirlingS1[n, n-k+1]; Flatten[Table[t[n, k], {n, 1, m}, {k, 1, n}]][[1 ;; len]] (* Jean-François Alcover, May 31 2011 *)
Flatten@Table[CoefficientList[Product[1-k x, {k, 1, n}], x], {n, 0, 8}] (* Oliver Seipel, Jun 14 2024 *)
Flatten@Table[Coefficient[Product[x-k, {k, 0, n-1}], x, Reverse@Range[n]], {n, Range[9]}] (* Oliver Seipel, Jun 14 2024, after  Ralf Stephan *)

T(n,k)=if(n<1,0,n!*polcoeff(binomial(x,n),n-k+1))

T(n,k)=if(n<1,0,n!*polcoeff(polcoeff(y*(1+y*x+x*O(x^n))^(1/y),n),k))

def T(n,k): return falling_factorial(x,n).expand().coefficient(x,n-k+1) # Ralf Stephan, Dec 11 2016

n!*binomial(x, n) = Sum_{k=1..n-1} T(n, k)*x^(n-k).
|A008276(n, k)| = T(n-1, k-1) where T(n, k) is the triangle, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...]; A008276(n, k) = T(n-1, k-1) where T(n, k) is the triangle, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, ...]. Here DELTA is the operator defined in A084938. - Philippe Deléham, Dec 30 2003
|T(n, k)| = Sum_{m=0..n} A008517(k, m+1)*binomial(n+m, 2*(k-1)), n >= k >= 1. A008517 is the second-order Eulerian triangle. See the Graham et al. reference p. 257, eq. (6.44).
A094638 formula for unsigned T(n, k).
|T(n, k)| = Sum_{m=0..min(k-1, n-k)} A112486(k-1, m)*binomial(n-1, k-1+m) if n >= k >= 1, else 0. - Wolfdieter Lang, Sep 12 2005, see A112486.
|T(n, k)| = (f(n-1, k-1)/(2*(k-1))!)* Sum_{m=0..min(k-1, n-k)} A112486(k-1, m)*f(2*(k-1), k-1-m)*f(n-k, m) if n >= k >= 1, else 0, where f(n, k) stands for the falling factorial n*(n-1)*...*(n-(k-1)) and f(n, 0):=1. - Wolfdieter Lang, Sep 12 2005, see A112486.
With P(n,t) = Sum_{k=0..n-1} T(n,k+1) * t^k = (1-t)*(1-2*t)*...*(1-(n-1)t) and P(0,t) = 1, exp(P(.,t)*x) = (1+t*x)^(1/t) . Compare A094638. T(n,k+1) = (1/k!) (D_t)^k (D_x)^n ( (1+t*x)^(1/t) - 1 ) evaluated at t=x=0 . - Tom Copeland, Dec 09 2007
Product_{i=1..n} (x-i) = Sum_{k=0..n} T(n,k)*x^k. - Reinhard Zumkeller, Dec 29 2007
E.g.f.: Sum_{n>=0} (Sum_{k=0..n} T(n,n-k)*t^k)/n!) = Sum_{n>=0} (x)n * t^k/n! = exp(x * log(1+t)), with (x)_n the n-th falling factorial polynomial. - _Ralf Stephan, Dec 11 2016
Sum_{j=0..m} T(m, m-j)*s2(j+k+1, m) =  m^k, where s2(j, m) are Stirling numbers of the second kind. - Tony Foster III, Jul 25 2019
For n>=2, Sum_{k=1..n} k*T(n,k) = (-1)^(n-1)*(n-2)!. - Zizheng Fang, Dec 27 2020

A000399 Unsigned Stirling numbers of first kind s(n,3).

Original entry on oeis.org

1, 6, 35, 225, 1624, 13132, 118124, 1172700, 12753576, 150917976, 1931559552, 26596717056, 392156797824, 6165817614720, 102992244837120, 1821602444624640, 34012249593822720, 668609730341153280, 13803759753640704000

Offset: 3

N. J. A. Sloane

Number of permutations of n elements with exactly 3 cycles.

The asymptotic expansion of the higher order exponential integral E(x,m=3,n=1) ~ exp(-x)/x^3*(1 - 6/x + 35/x^2 - 225/x^3 + 1624/x^4 - 13132/x^5 + ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009

			(-log(1-x))^3 = x^3 + 3/2*x^4 + 7/4*x^5 + 15/8*x^6 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
Shanzhen Gao, Permutations with Restricted Structure (in preparation). - Shanzhen Gao, Sep 14 2010
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Robert Israel, Table of n, a(n) for n = 3..412 (3..100 from T. D. Noe)
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].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 32.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv:1201.1323 [math.CO], 2012.
Sergey Kitaev and Jeffrey Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012), #12.4.7.

Cf. A000254, A000454, A000482, A001233, A001234, A008275, A243569, A243570.

A000399:=func< n | Abs(StirlingFirst(n, 3)) >; [ A000399(n): n in [3..25] ]; // Klaus Brockhaus, Jan 14 2011

seq(abs(Stirling1(n,3)),n=3..30); # Robert Israel, Jul 05 2015

a=Log[1/(1-x)];Range[0,20]! CoefficientList[Series[a^3/3!,{x,0,20}],x]
f[n_] := Abs@ StirlingS1[n, 3]; Array[f, 19, 3]
Abs[StirlingS1[Range[3,30],3]] (* Harvey P. Dale, Jun 23 2014 *)
f[n_] := Gamma[n]*(HarmonicNumber[n - 1]^2 + Zeta[2, n] - Zeta[2])/2; Array[f, 19, 3] (* Robert G. Wilson v, Jul 05 2015 *)

f := proc(n) option remember; begin n^3*f(n-3)-(3*n^2+3*n+1)*f(n-2)+3*(n+1)*f(n-1) end_proc: f(0) := 1: f(1) := 6: f(2) := 35:

for(n=2,50,print1(polcoeff(prod(i=1,n,x+i),2,x),","))

[stirling_number1(i+2,3) for i in range(1,22)] # Zerinvary Lajos, Jun 27 2008

Let P(n-1,X) = (X+1)(X+2)(X+3)...(X+n-1); then a(n) is the coefficient of X^2; or a(n) = P''(n-1,0)/2!. - Benoit Cloitre, May 09 2002 [Edited by Petros Hadjicostas, Jun 29 2020 to agree with the offset 3]
E.g.f.: -log(1-x)^3/3!.
a(n) is the coefficient of x^(n+3) in (-log(1-x))^3, multiplied by (n+3)!/6.
a(n) = ((Sum_{i=1..n-1} 1/i)^2 - Sum_{i=1..n-1} 1/i^2)*(n-1)!/2 for n >= 3. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000
a(n) = det(|S(i+3,j+2)|, 1 <= i,j <= n-3), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
a(n) = Gamma(n)*(HarmonicNumber(n-1)^2 + Zeta(2,n) - Zeta(2))/2. - Gerry Martens, Jul 05 2015
From Petros Hadjicostas, Jun 28 2020: (Start)
a(n) = (n-3)! + (2*n-3)*a(n-1) - (n-2)^2*a(n-2) for n >= 5.
a(n) = 3*(n-2)*a(n-1) - (3*n^2-15*n+19)*a(n-2) + (n-3)^3*a(n-3) for n >= 6. (End)

A001711 Generalized Stirling numbers.

Original entry on oeis.org

1, 7, 47, 342, 2754, 24552, 241128, 2592720, 30334320, 383970240, 5231113920, 76349105280, 1188825724800, 19675048780800, 344937224217600, 6386713749964800, 124548748102195200, 2551797512248320000, 54804198761303040000, 1231237843834521600000

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=3) ~ exp(-x)/x^2*(1 - 7/x + 47/x^2 - 342/x^3 + 2754/x^4 - 24552/x^5 + 241128/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009
For n > 4, a(n) mod n = 0 for n composite, = n-3 for n prime. - Gary Detlefs, Jul 18 2011
From Petros Hadjicostas, Jun 11 2020: (Start)
For nonnegative integers n, m and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) using slightly different notation. They were further examined by Mitrinovic and Mitrinovic (1962).
These numbers are defined via the g.f. Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0.
As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m. (Because an empty product is by definition 1, we may let R_0^0(a,b) = 1.)
With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m). (Array A008275 is the same as array A048994 but with no zero row and no zero column.)
We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.
For the current sequence, a(n) = R_{n+1}^1(a=-3, b=-1) for n >= 0. (End)

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..440[Terms 0 to 100 computed by T. D. Noe; terms 101 to 440 computed by G. C. Greubel, Jan 15 2017]
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356. [The numbers R_n^m(a,b) are introduced.]
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49. [Annotated scanned copy]
J. Riordan, Letter of 04/11/74.

Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k=2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564.

a := n-> add(1/2*((n+3)!/(k+3)), k=0..n): seq(a(n), n=0..19); # Zerinvary Lajos, Jan 22 2008
a := n -> (n+1)!*hs2(n+1): hs2 := n-> add(hs(k), k=0..n): hs := n-> add(h(k), k=0..n): h := n-> add(1/k, k=1..n): seq(a(n), n=0..19); # Gary Detlefs, Jan 01 2011

f[k_] := k + 2; t[n_] := Table[f[k], {k, 1, n}]; a[n_] := SymmetricPolynomial[n - 1, t[n]]; Table[a[n], {n, 1, 16}]; (* Clark Kimberling, Dec 29 2011 *)
Table[(n + 3)!*Sum[1/(2*k + 4), {k, 1, n + 1}], {n,0,100}] (* G. C. Greubel, Jan 15 2017 *)

for(n=0, 19, print1((n+1)! * sum(k=0, n, binomial(k + 2, 2) / (n + 1 - k)),", ")) \\ Indranil Ghosh, Mar 13 2017

R(n,m,a,b) =  sum(k=0, n-m, (-1)^k*a^k*b^(n-m-k)*binomial(m+k,k)*stirling(n, m+k,1));
aa(n) = R(n+1,1,-3,-1);
for(n=0, 19, print1(aa(n), ",")) \\ Petros Hadjicostas, Jun 11 2020

E.g.f.: -log(1 - x)/(1 - x)^3 if offset 1. With offset 0: (d/dx)(-log(1 - x)/(1 - x)^3) = (1 - 3*log(1 - x))/(1 - x)^4.
a(n) = Sum_{k=0..n} ((-1)^(n+k)*(k+1)*3^k*Stirling1(n+1, k+1)). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
a(n) = n!*Sum_{k=0..n-1} ((-1)^k*binomial(-3,k)/(n-k)). - Milan Janjic, Dec 14 2008
a(n) = ( A000254(n+3) - 3*A001710(n+3) )/2. - Gary Detlefs, May 24 2010
a(n) = ((n+3)!/4) * (2*h(n+3) - 3), where h(n) = Sum_{k=1..n} (1/k) is the n-th harmonic number. - Gary Detlefs, Aug 15 2010
a(n) = n!*[2]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n. With offset 1. - Gary Detlefs, Jan 04 2011
a(n) = (n+3)! * Sum_{k=1..n+1} (1/(2*k+4)). - Gary Detlefs, Sep 14 2011
a(n) = (n+1)! * Sum_{k=0..n} (binomial(k+2,2)/(n+1-k)). - Gary Detlefs, Dec 01 2011
a(n) = A001705(n+2) - A182541(n+4). - Anton Zakharov, Jul 02 2016
a(n) ~ n^(n+7/2) * exp(-n) * sqrt(Pi/2) * log(n) * (1 + (gamma - 3/2)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 12 2016
Conjectural D-finite with recurrence: a(n) + (-2*n-5)*a(n-1) + (n+2)^2*a(n-2)=0. - R. J. Mathar, Feb 16 2020
From Petros Hadjicostas, Jun 11 2020: (Start)
Since a(n) = R_{n+1}^1(a=-3, b=-1), it follows from Mitrinovic (1961) and Mitrinovic and Mitrinovic (1962) that:
a(n) = [x] Product_{r=0}^n (x + 3 + r) = (Product_{r=0}^n (3 + r)) * Sum_{s=0}^n 1/(3 + s).
a(n) = (n + 2)!/2 + (n + 3)*a(n-1) for n >= 1. [This can be used to prove R. J. Mathar's recurrence above.] (End)

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

Maple programs corrected and edited by Johannes W. Meijer, Nov 28 2012

A112007 Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.

Original entry on oeis.org

1, 2, 1, 6, 8, 1, 24, 58, 22, 1, 120, 444, 328, 52, 1, 720, 3708, 4400, 1452, 114, 1, 5040, 33984, 58140, 32120, 5610, 240, 1, 40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 362880, 3733920, 11026296, 12440064, 5765500, 1062500, 67260, 1004, 1

Offset: 0

Wolfdieter Lang, Sep 12 2005

This is the row reversed second-order Eulerian triangle A008517(k+1,k+1-m). For references see A008517.
The o.g.f. for the k-th diagonal, k >= 1, of the unsigned Stirling1 triangle |A008275| is G1(1,x)=1/(1-x) if k=1 and G1(k,x) = g1(k-2,x)/(1-x)^(2*k-1), if k >= 2, with the row polynomials g1(k;x):=Sum_{m=0..k} a(k,m)*x^m.
The recurrence eq. for the row polynomials is g1(k,x)=((k+1)+k*x)*g1(k-1,x) + x*(1-x)*(d/dx)g1(k-1,x), k >= 1, with input g1(0,x):=1.
The column sequences start with A000142 (factorials), A002538, A002539, A112008, A112485.
This o.g.f. computation was inspired by Bender et al. article where the Stirling polynomials have been rediscussed.
The A163936 triangle is identical to the triangle given above except for an extra right hand column [1, 0, 0, 0, ... ]. The A163936 triangle is related to the higher order exponential integrals E(x,m,n), see A163931 and A163932. - Johannes W. Meijer, Oct 16 2009

			Triangle begins:
    1;
    2,   1;
    6,   8,   1;
   24,  58,  22,   1;
  120, 444, 328,  52,   1;
  ...
G.f. for k=3 sequence A000914(n-1), [2,11,35,85,175,322,546,...], is G1(3,x)= g1(1,x)/(1-x)^5= (2+x)/(1-x)^5.

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
Peter Bala, Diagonals of triangles with generating function exp(t*F(x))
C. M. Bender, D. C. Brody and B. K. Meister, Bernoulli-like polynomials associated with Stirling Numbers, arXiv:math-ph/0509008 [math-ph], 2005.
E. Burlachenko, Composition polynomials of the RNA matrix and B-composition polynomials of the Riordan pseudo-involution, arXiv:1907.12272 [math.NT], 2019.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
Wolfdieter Lang, First 10 rows.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239. See 234. [_Tom Copeland_, Jan 03 2016]
Andrew Elvey Price, 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.

Row sums give A001147(k+1) = (2*k+1)!!, k>=0.

a:= proc(k,m) option remember; if m >= 0 and k >= 0 then (k+m+1)*procname(k-1,m)+(k-m+1)*procname(k-1,m-1) else 0 fi end proc:
a(0,0):= 1:
seq(seq(a(k,m),m=0..k),k=0..10); # Robert Israel, Jul 20 2017

a[k_, m_] = Sum[(-1)^(k + n + 1)*Binomial[2k + 3, n]*StirlingS1[m + k - n + 2, m + 1 - n], {n, 0, m}]; Flatten[Table[a[k, m], {k, 0, 8}, {m, 0, k}]][[1 ;; 45]] (* Jean-François Alcover, Jun 01 2011, after Johannes W. Meijer *)

a(k, m)=sum(n=0, m, (-1)^(k + n + 1)*binomial(2*k + 3, n)*stirling(m + k - n + 2, m + 1 - n, 1));
for(k=0, 10, for(m=0, k, print1(a(k, m),", "))) \\ Indranil Ghosh, Jul 21 2017

a(k, m) = (k+m+1)*a(k-1, m) + (k-m+1)*a(k-1, m-1), if k >= m >= 0, a(0, 0)=1; a(k, -1):=0, otherwise 0.
a(k,m) = Sum_{n=0..m} (-1)^(k+n+1)*C(2*k+3,n)*Stirling1(m+k-n+2,m+1-n). - Johannes W. Meijer, Oct 16 2009
The compositional inverse (with respect to x) of y = y(t,x) = (x+t*log(1-x)) is  x = x(t,y) = 1/(1-t)*y + t/(1-t)^3*y^2/2! + (2*t+t^2)/(1-t)^5*y^3/3! + (6*t+8*t^2+t^3)/(1-t)^7*y^4/4! + .... The numerator polynomials of the rational functions in t are the row polynomials of this triangle. As observed above, the rational functions in t are the generating functions for the diagonals of |A008275|. See the Bala link for a proof. Cf. A008517. - Peter Bala, Dec 02 2011

A094816 Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 6, 1, 1, 24, 29, 10, 1, 1, 89, 145, 75, 15, 1, 1, 415, 814, 545, 160, 21, 1, 1, 2372, 5243, 4179, 1575, 301, 28, 1, 1, 16072, 38618, 34860, 15659, 3836, 518, 36, 1, 1, 125673, 321690, 318926, 163191, 47775, 8274, 834, 45, 1, 1, 1112083, 2995011

Offset: 0

Philippe Deléham, Jun 12 2004

The a-sequence for this Sheffer matrix is A027641(n)/A027642(n) (Bernoulli numbers) and the z-sequence is A130189(n)/ A130190(n). See the W. Lang link.
Take the lower triangular matrix in A049020 and invert it, then read by rows! - N. J. A. Sloane, Feb 07 2009
Exponential Riordan array [exp(x), log(1/(1-x))]. Equal to A007318*A132393. - Paul Barry, Apr 23 2009
A signed version of the triangle appears in [Gessel]. - Peter Bala, Aug 31 2012
T(n,k) is the number of permutations over all subsets of {1,2,...,n} (Cf. A000522) that have exactly k cycles.  T(3,2) = 6: We permute the elements of the subsets {1,2}, {1,3}, {2,3}. Each has one permutation with 2 cycles.  We permute the elements of {1,2,3} and there are three permutations that have 2 cycles. 3*1 + 1*3 = 6. - Geoffrey Critzer, Feb 24 2013
From Wolfdieter Lang, Jul 28 2017: (Start)
In Chihara's book the row polynomials (with rising powers) are the Charlier polynomials (-1)^n*C^(a)_n(-x), with a = -1, n >= 0. See  p. 170, eq. (1.4).
In Ismail's book the present Charlier polynomials are denoted by C_n(-x;a=1) on p. 177, eq. (6.1.25). (End)
The triangle T(n,k) is a representative of the parametric family of triangles T(m,n,k), whose columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=-1. See A381082. - Igor Victorovich Statsenko, Feb 14 2025

			From _Paul Barry_, Apr 23 2009: (Start)
Triangle begins
  1;
  1,     1;
  1,     3,     1;
  1,     8,     6,     1;
  1,    24,    29,    10,     1;
  1,    89,   145,    75,    15,    1;
  1,   415,   814,   545,   160,   21,   1;
  1,  2372,  5243,  4179,  1575,  301,  28,  1;
  1, 16072, 38618, 34860, 15659, 3836, 518, 36, 1;
Production matrix is
  1, 1;
  0, 2, 1;
  0, 1, 3,  1;
  0, 1, 3,  4,  1;
  0, 1, 4,  6,  5,  1;
  0, 1, 5, 10, 10,  6,  1;
  0, 1, 6, 15, 20, 15,  7,  1;
  0, 1, 7, 21, 35, 35, 21,  8, 1;
  0, 1, 8, 28, 56, 70, 56, 28, 9, 1; (End)

T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, London, Paris, 1978, Ch. VI, 1., pp. 170-172.
Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005, EMA, Vol. 98, p. 177.

Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 6.
Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 8.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 22.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011) 11.6.7.
Ira Gessel, Congruences for Bell and Tangent numbers, The Fibonacci Quarterly, Vol. 19, Number 2, 1981.
Aoife Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First 10 rows and more.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.

Columns k=0..4 give A000012, A002104, A381021, A381022, A381023.
Diagonals: A000012, A000217.
Row sums A000522, alternating row sums A024000.
KummerU(-n,1-n-x,z): this sequence (z=1), |A137346| (z=2), A327997 (z=3).

A094816 := (n,k) -> (-1)^(n-k)*add(binomial(-j-1,-n-1)*Stirling1(j,k), j=0..n):
seq(seq(A094816(n, k), k=0..n), n=0..9); # Peter Luschny, Apr 10 2016

nn=10;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[Series[ Exp[x]/(1-x)^y,{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Feb 24 2013 *)
Flatten[Table[(-1)^(n-k) Sum[Binomial[-j-1,-n-1] StirlingS1[j,k],{j,0,n}], {n,0,9},{k,0,n}]] (* Peter Luschny, Apr 10 2016 *)
p[n_] := HypergeometricU[-n, 1 - n - x, 1];
Table[CoefficientList[p[n], x], {n,0,9}] // Flatten (* Peter Luschny, Oct 27 2019 *)

{T(n, k)= local(A); if( k<0 || k>n, 0, A = x * O(x^n); polcoeff( n! * polcoeff( exp(x + A) / (1 - x + A)^y, n), k))} /* Michael Somos, Nov 19 2006 */

def a_row(n):
    s = sum(binomial(n,k)*rising_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019

E.g.f.: exp(t)/(1-t)^x = Sum_{n>=0} C(x,n)*t^n/n!.
Sum_{k = 0..n} T(n, k)*x^k = C(x, n), Charlier polynomials; C(x, n)= A024000(n), A000012(n), A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n), A095722(n), A095740(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Feb 27 2013
T(n+1, k) = (n+1)*T(n, k) + T(n, k-1) - n*T(n-1, k) with T(0, 0) = 1, T(0, k) = 0 if k>0, T(n, k) = 0 if k<0.
PS*A008275*PS as infinite lower triangular matrices, where PS is a triangle with PS(n, k) = (-1)^k*A007318(n, k). PS = 1/PS. - Gerald McGarvey, Aug 20 2009
T(n,k) = (-1)^(n-k)*Sum_{j=0..n} C(-j-1, -n-1)*S1(j, k) where S1 are the signed Stirling numbers of the first kind. - Peter Luschny, Apr 10 2016
Absolute values T(n,k) of triangle (-1)^(n+k) T(n,k) where row n gives coefficients of x^k, 0 <= k <= n, in expansion of Sum_{k=0..n} binomial(n,k) (-1)^(n-k) x^{(k)}, where x^{(k)} := Product_{i=0..k-1} (x-i), k >= 1, and x^{(0)} := 1, the falling factorial powers. - Daniel Forgues, Oct 13 2019
From Peter Bala, Oct 23 2019: (Start)
The n-th row polynomial is
   R(n, x) = Sum_{k = 0..n} (-1)^k*binomial(n, k)*k! * binomial(-x, k).
These polynomials occur in series acceleration formulas for the constant
   1/e = n! * Sum_{k >= 0} (-1)^k/(k!*R(n,k)*R(n,k+1)), n >= 0. (cf. A068985, A094816 and A137346). (End)
R(n, x) = KummerU[-n, 1 - n - x, 1]. - Peter Luschny, Oct 27 2019
Sum_{j=0..m} (-1)^(m-j) * Bell(n+j) * T(m,j) = m! * Sum_{k=0..n} binomial(k,m) * Stirling2(n,k). - Vaclav Kotesovec, Aug 06 2021
From Natalia L. Skirrow, Jun 11 2025: (Start)
G.f.: 2F0(1,y;x/(1-x)) / (1-x).
Polynomial for the n-th row is R(n,y) = 2F0(-n,y;-1).
Falling g.f. for n-th row: Sum_{k=0..n} a(n,k)*(y)_k = [x^0] 2F0(1,-n;-1/x) * (1+log(1/(1-x)))^y = [x^n] e^x * Gamma(n+1,x) * (1+log(1/(1-x)))^y, where (y)_k = y!/(y-k)! denotes the falling factorial. (End)

A002657 Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).

Original entry on oeis.org

1, 1, 5, 9, 251, 475, 19087, 36799, 1070017, 2082753, 134211265, 262747265, 703604254357, 1382741929621, 8164168737599, 5362709743125, 8092989203533249, 15980174332775873, 12600467236042756559

Offset: 0

N. J. A. Sloane

These coefficients (with alternating signs) are also known as the Nørlund [or Norlund, Noerlund or Nörlund] numbers. [After the Danish mathematician Niels Erik Nørlund (1885-1981). - Amiram Eldar, Jun 17 2021]
The denominators are found in A002790. The alternating rational sequence ((-1)^n)*a(n)/A002790(n)is the z-sequence for the Stirling2 triangle A008277(n+1,k+1), n>=k>=0. This is the Sheffer (exp(x),exp(x)-1) triangle. See the W. Lang link under A006232 for Sheffer a- and z-sequences with references, and the conversion to S. Roman's notation. The a-sequence is A006232(n)/A006233(n). - Wolfdieter Lang, Oct 06 2011 [This is the Sheffer triangle A007318*A048993. Added Jun 20 2017]
A simple series with the signless Cauchy numbers of second type C2(n) leads to Euler's constant: gamma = 1 - Sum_{n >=1} C2(n)/(n*(n+1)!) = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A075266 and A262235. - Iaroslav V. Blagouchine, Sep 15 2015

			1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
Louis Melville Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Ibrahim M. Alabdulmohsin, The Language of Finite Differences, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, 2018, pp. 133-149.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
Donghyun Kim and Jaeseong Oh, Extending the science fiction and the Loehr--Warrington formula, arXiv:2409.01041 [math.CO], 2024. See p. 32.
Takao Komatsu, Convolution Identities for Cauchy Numbers of the Second Kind, Kyushu Journal of Mathematics, Vol. 69, No. 1 (2015), pp. 125-144.
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., Vol. 45, No. 2 (2007), pp. 133-137.
Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq., Vol. 17 (2014), Article 14.4.6.
Rui-Li Liu and Feng-Zhen Zhao, Log-concavity of two sequences related to Cauchy numbers of two kinds, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.
Louis Melville Milne-Thompson, Calculus of Finite Differences, 1951. [Annotated scan of pages 135, 136 only]
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer, 1924, p. 461.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 461 [Annotated scanned copy of pages 144-151 and 456-463]
Michael O. Rubinstein, Identities for the Riemann zeta function, Ramanujan J., Vol. 27, No. 1 (2012), pp. 29-42; arXiv preprint, arXiv:0812.2592 [math.NT], 2008-2009.
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., Vol. 309, No. 12 (2009), pp. 3830-3842.
Index entries for sequences related to Bernoulli numbers.

Cf. A002206, A002207, A002208, A002209, A002790, A006232, A006233, A075266, A075267, A262235.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-x/((1-x)*Log(1-x)) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Oct 29 2018

seq(numer(add((-1)^(n-k)*Stirling1(n,k)/(k+1),k=0..n)),n=0..10); # Peter Luschny, Apr 28 2009

Table[Abs[Numerator[NorlundB[n,n]]],{n,0,30}](* Vladimir Joseph Stephan Orlovsky, Dec 30 2010 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ NorlundB[ n, n]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator@Integrate[ Pochhammer[ x, n], {x, 0, 1}]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator[ n! SeriesCoefficient[ -x / ((1 - x) Log[ 1 - x]), {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator[ n! SeriesCoefficient[ (x / (Exp[x] - 1))^n, {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)

v(n):=if n=0 then 1 else 1-sum(v(i)/(n-i+1),i,0,n-1);
makelist(num(n!*v(n)),n,0,10); /* Vladimir Kruchinin, Aug 28 2013 */

Numerator of integral of x(x+1)...(x+n-1) from 0 to 1.
E.g.f.: -x/((1-x)*log(1-x)). (Note: the numerator of the coefficient of x^n/n! is a(n). - Michael Somos, Jul 12 2014). E.g.f. rewritten by Iaroslav V. Blagouchine, May 07 2016
Numerator of Sum_{k=0..n} (-1)^(n-k) A008275(n,k)/(k+1). - Peter Luschny, Apr 28 2009
a(n) = numerator(n!*v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013

A048594 Triangle T(n,k) = k! * Stirling1(n,k), 1<=k<=n.

Original entry on oeis.org

1, -1, 2, 2, -6, 6, -6, 22, -36, 24, 24, -100, 210, -240, 120, -120, 548, -1350, 2040, -1800, 720, 720, -3528, 9744, -17640, 21000, -15120, 5040, -5040, 26136, -78792, 162456, -235200, 231840, -141120, 40320, 40320, -219168, 708744, -1614816, 2693880, -3265920, 2751840, -1451520, 362880

Offset: 1

Oleg Marichev (oleg(AT)wolfram.com)

Row sums (unsigned) give A007840(n), n>=1; (signed): A006252(n), n>=1.

Apart from signs, coefficients in expansion of n-th derivative of 1/log(x).

			Triangle begins
   1;
  -1,    2;
   2,   -6,   6;
  -6,   22, -36,   24;
  24, -100, 210, -240, 120; ...
The 2nd derivative of 1/log(x) is -2/x^3*log(x)^2 - 6/x^3*log(x)^3 - 6/x^3*log(x)^4.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Stirling Number of the First Kind
Wikipedia, Stirling numbers and exponential generating functions

Cf. A008275, A019538, A075181.
Cf. A133942 (left edge), A000142 (right edge), A006252 (row sums), A238685 (central terms).
Row sums: A007840 (unsigned), A006252 (signed).

a048594 n k = a048594_tabl !! (n-1) !! (k-1)
a048594_row n = a048594_tabl !! (n-1)
a048594_tabl = map snd $ iterate f (1, [1]) where
   f (i, xs) = (i + 1, zipWith (-) (zipWith (*) [1..] ([0] ++ xs))
                                   (map (* i) (xs ++ [0])))
-- Reinhard Zumkeller, Mar 02 2014

/* As triangle: */ [[Factorial(k)*StirlingFirst(n,k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Dec 15 2015

with(combinat): A048594 := (n,k)->k!*stirling1(n,k);

Flatten[Table[k!*StirlingS1[n,k], {n,10}, {k,n}]] (* Harvey P. Dale, Aug 28 2011 *)
Join @@ CoefficientRules[ -Table[ D[ 1/Log[z], {z, n}], {n, 9}] /. Log[z] -> -Log[z], {1/z, 1/Log[z]}, "NegativeLexicographic"][[All, All, 2]] (* Oleg Marichev (oleg(AT)wolfram.com) and Maxim Rytin (m.r(AT)inbox.ru); submitted by Robert G. Wilson v, Aug 29 2011 *)

{T(n, k)= if(k<1 || k>n, 0, stirling(n, k)* k!)} /* Michael Somos Apr 11 2007 */

def A048594(n,k): return (-1)^(n-k)*factorial(k)*stirling_number1(n,k)
flatten([[A048594(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 24 2023

T(n, k) = k*T(n-1, k-1) - (n-1)*T(n-1, k) if n>=k>=1, T(n, 0) = 0 and T(1, 1)=1, else 0.
E.g.f. k-th column: log(1+x)^k, k>=1.
From Peter Bala, Nov 25 2011: (Start):
E.g.f.: 1/(1-t*log(1+x)) = 1 + t*x + (-t+2*t^2)*x^2/2! + ....
The row polynomials are given by D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator exp(-x)*d/dx.
(End)

A002790 Denominators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 84, 24, 90, 20, 132, 24, 5460, 840, 360, 16, 1530, 180, 7980, 840, 13860, 440, 1656, 720, 81900, 6552, 216, 112, 3480, 240, 114576, 7392, 117810, 2380, 1260, 72, 3838380, 207480, 32760, 560, 568260, 27720, 238392, 55440, 869400, 2576, 236880

Offset: 0

N. J. A. Sloane

The numerators are given in A002657.
These coefficients (with alternating signs) are also known as the Nørlund [or Norlund, Noerlund or Nörlund] numbers.
A simple series with the signless Cauchy numbers of second type C2(n) leads to Euler's constant: gamma = 1 - Sum_{n >=1} C2(n)/(n*(n+1)!) = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A075266 and A262235. - Iaroslav V. Blagouchine, Sep 15 2015
a(n) appears to be divisible by n+1. - Hal M. Switkay, Aug 15 2025

			1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Ibrahim M. Alabdulmohsin, The Language of Finite Differences, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, pp 133-149.
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
C. H. Karlson & N. J. A. Sloane, Correspondence, 1974
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., 45 (2007), 133-137.
Guo-Dong Liu, H. M. Srivastava, Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq. 17 (2014) # 14.4.6.
Rui-Li Liu and Feng-Zhen Zhao, Log-concavity of two sequences related to Cauchy numbers of two kinds, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math. 306 (2006), no. 16, 1906-1920.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951. [Annotated scan of pages 135, 136 only]
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 461.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 461 [Annotated scanned copy of pages 144-151 and 456-463]
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., 309 (2009), 3830-3842.
Index entries for sequences related to Bernoulli numbers.

Cf. A002657, A075266, A075267, A262235.

See also A002208, A002209, A002206, A002207, A006232, A006233.

m:=60; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-x/((1-x)*Log(1-x)) )); [Denominator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Oct 28 2018

A002790 := proc(n)
    denom(add((-1)^k*Stirling1(n, k)/(k+1), k=0..n)) ;
end proc: # Peter Luschny, Apr 28 2009
v := proc(n) option remember; ifelse(n=0, 1, 1 - add(v(i)/(n-i+1), i=0..n-1)) end:
seq(denom(n!*v(n)), n = 0..46); # after Vladimir Kruchinin, Peter Luschny, Aug 17 2025

Table[ Denominator[ NorlundB[n, n]], {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Dec 30 2010 *)

v(n):=if n=0 then 1 else 1-sum(v(i)/(n-i+1),i,0,n-1);
makelist(denom(n!*v(n)),n,0,10); /* Vladimir Kruchinin, Aug 28 2013 */

Denominator of integral of x(x+1)...(x+n-1) from 0 to 1.
E.g.f.: -x/((1-x)*log(1-x)). - Corrected by Iaroslav V. Blagouchine, May 07 2016.
Denominator of Sum_{k=0..n} (-1)^k A008275(n,k)/(k+1). - Peter Luschny, Apr 28 2009
a(n) = denominator(n!*v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013

cp's OEIS Frontend

A000914 Stirling numbers of the first kind: s(n+2, n).

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A132440 Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.

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A008276 Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.

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A000399 Unsigned Stirling numbers of first kind s(n,3).

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A001711 Generalized Stirling numbers.

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