A000262 -id:A000262 - cp's OEIS frontend

A229263 E.g.f.: Product_{n>=1} cosh(x^n).

1, 1, 13, 541, 32761, 3782521, 570649861, 126354119893, 34059666142321, 12697511966492401, 5418397453551516541, 2950382131846118771341, 1848796902719228099999593, 1394126061848631877574788201, 1187817128863650862040235107701, 1196980698779612997551160117313861

Offset: 0

Paul D. Hanna, Sep 18 2013

nonn

			E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 32761*x^8/8! +...
where
A(x) = cosh(x)*cosh(x^2)*cosh(x^3)*cosh(x^4)*cosh(x^5)*cosh(x^6)*...

Vaclav Kotesovec, Table of n, a(n) for n = 0..220

Cf. A000262, A270294, A270662, A270663.

{a(n)=(2*n)!*polcoeff(prod(m=1,n,cosh(x^m +x*O(x^(2*n)))),2*n)}
for(n=0,20,print1(a(n),", "))

A255807 E.g.f.: exp(Sum_{k>=1} k^2 * x^k).

Original entry on oeis.org

1, 1, 9, 79, 841, 10821, 162601, 2777419, 52960209, 1112813641, 25509407401, 632772511911, 16870674740569, 480717000225229, 14568646143888201, 467640968478534691, 15841420612530533281, 564519727866573515409, 21102817266052772063689, 825435163723385398719871

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if e.g.f. = exp(Sum_{k>=1} k^m * x^k) and m>0, then a(n) ~ (m+2)^(-1/2) * Gamma(m+2)^(1/(2*m+4)) * exp((m+2)/(m+1) * Gamma(m+2)^(1/(m+2)) * n^((m+1)/(m+2)) + zeta(-m) - n) * n^(n - 1/(2*m+4)).
It appears that the sequence a(n) taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k: if true, then the sequence a(n) taken modulo k would be periodic with period dividing k. - Peter Bala, Nov 14 2017
The above conjecture is true - see the Bala link. - Peter Bala, Jan 20 2018

G. C. Greubel, Table of n, a(n) for n = 0..250
P. Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A082579, A255819, A000262.

nmax=20; CoefficientList[Series[Exp[Sum[k^2*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k^2*x^k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 21 2016 *)

E.g.f.: exp(x*(1+x)/(1-x)^3).
a(n) ~ 2^(-7/8) * 3^(1/8) * n^(n-1/8) * exp(2^(9/4) * 3^(-3/4) * n^(3/4) - n).
a(n) = n!*y(n) where y(0)=1 and y(n)=(Sum_{k=0..n-1} (n-k)^3*y(k))/n for n>=1. - Benedict W. J. Irwin, Jun 02 2016
a(n) = (4*n-3)*a(n-1) - 2*(n-1)*(3*n-8)*a(n-2) + (n-1)*(n-2)*(4*n-11)*a(n-3) - (n-1)*(n-2)*(n-3)*(n-4)*a(n-4). - Peter Bala, Nov 12 2017
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(J_3(k)/k), where J_3() is the Jordan function (A059376). - Ilya Gutkovskiy, May 25 2019

A270663 E.g.f.: Product_{k>=1} cos(x^k) [even terms only].

Original entry on oeis.org

1, -1, -11, -181, -9239, -148681, -49402979, 6471717251, 42662277841, 658656817939439, 133531458273294661, 168943525289665105979, 19015164932231993967289, 62294481438650615377602599, 18546969159687034895328945901, 27398539855607539080934584895859

Offset: 0

Vaclav Kotesovec, Mar 21 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..224

Cf. A000262, A229263, A270294, A270662.

nmax = 20; Table[(Range[0, 2*nmax]! * CoefficientList[Series[Product[Cos[x^k], {k, 1, 2*nmax}], {x, 0, 2*nmax}], x])[[2*n + 1]], {n, 0, nmax}]

A293135 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} x^(j*i)/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 12, 0, 1, 1, 3, 12, 48, 0, 1, 1, 3, 13, 72, 360, 0, 1, 1, 3, 13, 72, 480, 2880, 0, 1, 1, 3, 13, 73, 500, 3780, 25200, 0, 1, 1, 3, 13, 73, 500, 4020, 35280, 241920, 0, 1, 1, 3, 13, 73, 501, 4050, 37380, 372960, 2903040, 0

Offset: 0

Seiichi Manyama, Oct 01 2017

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   2,   3,   3,   3, ...
   0,  12,  12,  13,  13, ...
   0,  48,  72,  72,  73, ...
   0, 360, 480, 500, 500, ...

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

Columns k=0..5 give A000007, A088311, A293138, A293195, A293196, A293197.
Rows n=0 gives A000012.
Main diagonal gives A000262.
Cf. A293139.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
    end:
A:= (n, k)-> n!*b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Oct 02 2017

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1, k]/j!, {j, 0, Min[k, n/i]}]]];
A[n_, k_] := n! b[n, n, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

A036074 Expansion of e.g.f. exp((exp(p*x) - p - 1)/p + exp(x)) for p=4.

Original entry on oeis.org

1, 2, 9, 55, 412, 3619, 36333, 408888, 5080907, 68914023, 1011165446, 15935379409, 268125052373, 4792458452162, 90605469012877, 1805135197261131, 37775862401203916, 827992670793489263

Offset: 0

N. J. A. Sloane

nonn

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036076, ...

mx = 16; p = 4; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 4^k * BellB[k, 1/4] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

a(n):=sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n); /* Vladimir Kruchinin, Sep 14 2010 */

a(n) = sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n), n > 0. - Vladimir Kruchinin, Sep 14 2010
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=4. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (4*n/LambertW(4*n))^n * exp(n/LambertW(4*n) + (4*n/LambertW(4*n))^(1/4) - n - 5/4) / sqrt(1 + LambertW(4*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters

A049120 Row sums of triangle A049029.

Original entry on oeis.org

1, 6, 61, 871, 15996, 358891, 9509641, 290528316, 10051973371, 388433817091, 16579346005806, 774580047063901, 39313104018590221, 2153825039102763846, 126681355435102649161, 7961385691338995966371, 532402860878855993673036, 37746950872336992298209151

Offset: 1

Wolfdieter Lang

Generalized Bell numbers B(5,1;n).

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

Cf. Generalized Bell numbers B(m, 1, n): A049118 (m=3), A049119 (m=4), this sequence (m=5), A049412 (m=6).

With[{nn=20},CoefficientList[Series[Exp[-1+1/Surd[1-4x,4]]-1,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 10 2019 *)

E.g.f. exp(-1+1/(1-4*x)^(1/4))-1.
Representation of a(n) as the n-th moment of a positive function on positive half-axis (Stieltjes moment problem), in Maple notation: a(n)=int(x^n*exp(-1)*exp(-1/4*x)*(1/96*x*hypergeom([],[5/4, 3/2, 7/4, 2],1/1024*x)+ 1/8*4^(3/4)*x^(1/4)/Pi*2^(1/2)*GAMMA(3/4)*hypergeom([],[1/4, 1/2,3/4, 5/4],1/1024*x)+1/8*4^(1/2)*x^(1/2)/Pi^(1/2)*hypergeom([],[1/2, 3/4, 5/4,3/2],1/1024*x)+1/24*4^(1/4)*x^(3/4)/GAMMA(3/4)*hypergeom([],[3/4, 5/4, 3/2,7/4],1/1024*x))/x, x=0..infinity),n=1,2... . - Karol A. Penson, Dec 16 2007
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)^5*d/dx. Cf. A000110, A000262, A049118 and A049119. - Peter Bala, Nov 25 2011
a(n) = (1/e) * (-4)^n * n! * Sum_{k>=0} binomial(-k/4,n)/k!. - Seiichi Manyama, Jan 17 2025

A049376 Row sums of triangle A046089.

Original entry on oeis.org

1, 1, 4, 22, 154, 1306, 12976, 147484, 1883932, 26680924, 414468496, 7001104936, 127677078904, 2498712779512, 52209534323584, 1159559538626896, 27269218041047056, 676732851527182864, 17669429275516846912, 484087943980439097184, 13882791112964223876256

Offset: 0

Wolfdieter Lang

nonn

a(n) is the number of n-permutations where each cycle has two (not necessarily distinct) roots. Here a root means a designated element in a cycle. Cf. A000262 which gives the number of n-permutations with a single root in each cycle. Note that the order of designating the elements is not important. Cf. (A bijection from endofunctions to "doubly" rooted trees where the order of designating the roots is important) Miklos Bona, A Walk Through Combinatorics, World Scientific Publishing, 2006, page 216. - Geoffrey Critzer, May 17 2012.

			a(2) = 4 because we have: (1'')(2'');(1''2);(12'');(1'2') where the permutations are given in cycle notation and the two roots in each cycle are designated by a '.

Alois P. Heinz, Table of n, a(n) for n = 0..436
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=3 of A291709.
Cf. A000085, A000262, A046089, A136658.
Cf. A004211, A375173.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+1)!/2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017
a := proc(n) option remember; `if`(n < 3, [1, 1, 4][n + 1],
a(n-1)*(3*n-2) - a(n-2)*3*(n-1)*(n-2) + a(n-3)*(n-1)*(n-2)*(n-3)) end:
seq(a(n), n=0..20); # after Emanuele Munarini, Peter Luschny, Sep 09 2017

nn = 15;Drop[Range[0, nn]! CoefficientList[Series[Exp[x/(1 - x) + x^2/2/(1 - x)^2], {x, 0, nn}], x], 1]  (* Geoffrey Critzer, May 17 2012 *)

E.g.f.: exp(p(x)) with p(x) := x*(2-x)/(2*(1-x)^2) (E.g.f. first column of A046089).
Lah transform of A000085: a(n) = Sum_{k=0..n} n!/k!*binomial(n-1,k-1) * A000085(k). - Vladeta Jovovic, Oct 02 2003
a(n+3) - (3*n+7)*a(n+2) + 3*(n+1)*(n+2)*a(n+1) - n*(n+1)*(n+2)* a(n) = 0. - Emanuele Munarini, Sep 08 2017
a(n) ~ n^(n-1/6) / sqrt(3) * exp(-1/3 + n^(1/3)/2 + 3*n^(2/3)/2 - n). - Vaclav Kotesovec, Oct 23 2017
E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^3 dx )^n / n!, where the constant of integration is taken to be zero. - Paul D. Hanna, Apr 27 2019
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004211(k).
a(n) = (1/exp(1/2)) * (-1)^n * n! * Sum_{k>=0} binomial(-2*k,n)/(2^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A059110 Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 13, 21, 9, 1, 73, 136, 78, 16, 1, 501, 1045, 730, 210, 25, 1, 4051, 9276, 7515, 2720, 465, 36, 1, 37633, 93289, 85071, 36575, 8015, 903, 49, 1, 394353, 1047376, 1053724, 519456, 137270, 20048, 1596, 64, 1, 4596553, 12975561

Offset: 0

Vladeta Jovovic, Jan 04 2001

L'(n,i) are unsigned Lah numbers (cf. A008297): L'(n,i)=n!/i!*binomial(n-1,i-1) for i >= 1, L'(0,0)=1, L'(n,0)=0 for n>0. T(n,0)=A000262(n); T(n,2)=A052852(n). Row sums A052897.
Exponential Riordan array [e^(x/(1-x)),x/(1-x)]. - Paul Barry, Apr 28 2007
From Wolfdieter Lang, Jun 22 2017: (Start)
The inverse matrix T^(-1) is exponential Riordan (aka Sheffer) (e^(-x), x/(1+x)): T^(-1)(n, m) = (-1)^(n-m)*A271705(n, m).
The a- and z-sequences of this Sheffer (aka exponential Riordan) matrix are a = [1,1,repeat(0)] and z(n) = (-1)^(n+1)*A028310(n)/A000027(n-1) with e.g.f. ((1+x)/x)*(1-exp(-x)). For a- and z-sequences see a W. Lang link under A006232 with references. (End)

			The triangle T = A007318*A271703 starts:
n\m       0        1        2       3       4      5     6    7  8 9 ...
0:        1
1:        1        1
2:        3        4        1
3:       13       21        9       1
4:       73      136       78      16       1
5:      501     1045      730     210      25      1
6:     4051     9276     7515    2720     465     36     1
7:    37633    93289    85071   36575    8015    903    49    1
8:   394353  1047376  1053724  519456  137270  20048  1596   64  1
9:  4596553 12975561 14196708 7836276 2404206 427518 44436 2628 81 1
... reformatted. - _Wolfdieter Lang_, Jun 22 2017
E.g.f. for T(n, 2) = 1/2!*(x/(1-x))^2*e^(x/(x-1)) = 1*x^2/2 + 9*x^3/3! + 78*x^4/4! + 730*x^5/5! + 7515*x^6/6 + ...
From _Wolfdieter Lang_, Jun 22 2017: (Start)
The z-sequence starts: [1, 1/2, -2/3, 3/4, -4/5, 5/6, -6/7, 7/8, -8/9, ...
T recurrence: T(3, 0) = 3*(1*T(2,0) + (1/2)*T(2, 1) + (-2/3)*T(2 ,1)) = 3*(3 + (1/2)*4 - (2/3)) = 13; T(3, 1) = 3*(T(2, 0)/1 + T(2, 1)) = 3*(3 + 4) = 21.
Meixner type recurrence for R(2, x): (D - D^2)*(3 + 4*x + x^2) = 4 + 2*x - 2 = 2*(1 + x), (D = d/dx).
General Sheffer recurrence for R(2, x): (1+x)*(1 + 2*D + D^2)*(1 + x) = (1+x)*(1 + x + 2) = 3 + 4*x + x^2. (End)

Muniru A Asiru, Rows n=0..50 of triangle, flattened
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A007318, A008297, A052852, A052897, A271703, A271705.

Concatenation([1],Flat(List([1..10],n->List([0..n],m->Sum([0..n],i-> Factorial(n)/Factorial(i)*Binomial(n-1,i-1)*Binomial(i,m)))))); # Muniru A Asiru, Jul 25 2018

A059110:= func< n,k | n eq 0 select 1 else Factorial(n-1)*Binomial(n,k)*Evaluate(LaguerrePolynomial(n-1, 1-k), -1) >;
[A059110(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

Lprime := proc(n,i)
    if n = 0 and i = 0 then
        1;
    elif k = 0 then
        0 ;
    else
        n!/i!*binomial(n-1,i-1) ;
    end if;
end proc:
A059110 := proc(n,k)
    add(Lprime(n,i)*binomial(i,k),i=0..n) ;
end proc: # R. J. Mathar, Mar 15 2013

(* First program *)
lp[n_, i_] := Binomial[n-1, i-1]*n!/i!; lp[0, 0] = 1; t[n_, m_] := Sum[lp[n, i]*Binomial[i, m], {i, 0, n}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2013 *)
(* Second program *)
A059110[n_, k_]:= If[n==0, 1, (n-1)!*Binomial[n, k]*LaguerreL[n-1, 1-k, -1]];
Table[A059110[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

def A059110(n, k): return 1 if n==0 else factorial(n-1)*binomial(n, k)*gen_laguerre(n-1, 1-k, -1)
flatten([[A059110(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

E.g.f. for column m: (1/m!)*(x/(1-x))^m*e^(x/(x-1)), m >= 0.
From Wolfdieter Lang, Jun 22 2017: (Start)
E.g.f. for row polynomials in powers of x (e.g.f. of the triangle): exp(z/(1-z))* exp(x*z/(1-z)) (exponential Riordan).
Recurrence: T(n, 0) = Sum_{j=0} z(j)*T(n-1, j), n >= 1, with z(n) = (-1)^(n+1)*A028310(n), T(0, 0) = 1, T(n, m) = 0 n < m, T(n, m) = n*(T(n-1, m-1)/m + T(n-1, m)), n >= m >= 1 (from the z- and a-sequence, see a comment above).
Meixner type recurrence for the (monic) row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: Sum_{k=0..n-1} (-1)^k*D^(k+1)*R(n, x) = n*R(n-1, x), n >=1, R(0, x) = 1, with D = d/dx.
General Sheffer recurrence: R(n, x) = (x+1)*(1+D)^2*R(n-1, x), n >=1, R(0, x) = 1.
(End)
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of this entry; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 18 2018
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (n-1)!*binomial(n, k)*LaguerreL(n-1, 1-k, -1) with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = A052897(n). (End)

A066668 Signed row sums of A066667.

Original entry on oeis.org

1, 1, 1, -1, -19, -151, -1091, -7841, -56519, -396271, -2442439, -7701409, 145269541, 4833158329, 104056218421, 2002667085119, 37109187217649, 679877731030049, 12440309297451121, 227773259993414719, 4155839606711748061, 74724654677947488521, 1293162252850914402221

Offset: 0

Christian G. Bower, Dec 17 2001

sign

Numerators in exp(x/(x+1)) power series (signs are different). - Benoit Cloitre, Mar 13 2002

Determinant of n X n matrix M=[m(i,j)] where m(i,i)=i, m(i,j)=1 if i>j, m(i,j)=j-i if j>i. - Vladeta Jovovic, Jan 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A111884.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(x-1))/(1-x)^2)); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2018

a := n -> n!*hypergeom([1-n], [2], 1):
seq(simplify(a(n)), n=1..19); # Peter Luschny, Mar 30 2015

CoefficientList[Series[E^(x/(x-1))/(1-x)^2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 13 2014 *)
Table[Sum[-BellY[n+1, k, -Range[n+1]!], {k, n+1}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))/(1-x)^2)) \\ G. C. Greubel, May 15 2018

A066668 = lambda n: (-1)^n*hypergeometric([-n-1,-n-1,-n],[-n-1],-1)
[Integer(A066668(n).n(100)) for n in range(23)] # Peter Luschny, Sep 22 2014

a(n) = n!LaguerreL(n, 1, 1). - Paul Barry, Sep 08 2004
E.g.f.: exp(x/(x-1))/(1-x)^2.
Conjecture: a(n) +(-2*n+1)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
E.g.f. with a different offset: 1 - product {n >= 1} (1 - x^n)^(phi(n)/n) = x + x^2/2 + x^3/6 - x^4/24 - 19*x^5/120 - ..., where phi(n) = A000010(n) is the Euler totient function. Cf. A000262. - Peter Bala, Jan 01 2014
a(n) = (-1)^n*hypergeom([-n-1,-n-1,-n],[-n-1],-1). - Peter Luschny, Sep 22 2014
a(n) = n!*hypergeom([1-n], [2], 1). - Peter Luschny, Mar 30 2015

A129652 Exponential Riordan array [e^(x/(1-x)),x].

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 73, 52, 18, 4, 1, 501, 365, 130, 30, 5, 1, 4051, 3006, 1095, 260, 45, 6, 1, 37633, 28357, 10521, 2555, 455, 63, 7, 1, 394353, 301064, 113428, 28056, 5110, 728, 84, 8, 1, 4596553, 3549177, 1354788, 340284, 63126, 9198, 1092, 108, 9, 1

Offset: 0

Paul Barry, Apr 26 2007

Satisfies the equation e^[x/(1-x),x] = e*[e^(x/(1-x)),x].
Row sums are A052844.
Antidiagonal sums are A129653.

			Triangle begins:
     1;
     1,    1;
     3,    2,    1;
    13,    9,    3,   1;
    73,   52,   18,   4,  1;
   501,  365,  130,  30,  5, 1;
  4051, 3006, 1095, 260, 45, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
T.-X. He, A symbolic operator approach to power series transformation-expansion formulas, JIS 11 (2008) 08.2.7.

Cf. A000262 (column 0), A052844 (row sums).
T(2n,n) gives A350461.
Cf. A052845, A111884, A133314.

A129652 := (n, k) -> (-1)^(k-n+1)*binomial(n,k)*KummerU(k-n+1, 2, -1);
seq(seq(round(evalf(A129652(n,k),99)),k=0..n),n=0..9); # Peter Luschny, Sep 17 2014
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1$2], add((p-> p+
     [0, p[1]*x^j])(b(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/i!, i=0..n))(b(n)[2]):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 21 2022

T[n_, k_] := If[k==n, 1, n!/k! Sum[Binomial[n-k-1, j]/(j+1)!, {j, 0, n-k-1}]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 14 2019 *)

Number triangle T(n,k)=(n!/k!)*sum{i=0..n-k, C(n-k-1,i)/(n-k-i)!}
From Peter Bala, May 14 2012 : (Start)
Array is exp(S*(I-S)^(-1)) where S is A132440 the infinitesimal generator for Pascal's triangle.
Column 0 is A000262.
T(n,k) = binomial(n,k)*A000262(n-k).
So T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then arrange the remaining n-k elements into a set of lists. (End)
T(n,k) = (-1)^(k-n+1)*C(n,k)*KummerU(k-n+1, 2, -1). - Peter Luschny, Sep 17 2014
From Tom Copeland, Mar 11 2016: (Start)
The row polynomials P_n(x) form an Appell sequence with e.g.f. e^(t*P.(x)) = e^[t / (1-t)] e^(x*t), so the lowering and raising operators are L = d/dx = D and the R = x + 1 / (1-D)^2 = x + 1 + 2 D + 3 D^2 + ..., satisfying L P_n(x) = n * P_(n-1)(x) and R P_n(x) = P_(n+1)(x).
(P.(x) + y)^n = Sum_{k=0..n} binomial(n,k) P_k(x) * y^(n-k) = P_n(x+y).
The Appell polynomial umbral compositional inverse sequence has the e.g.f. e^(t*Q.(x)) = e^[-t / (1-t)] e^(x*t) (see A111884 and A133314), so Q_n(P.(x)) = P_n(Q.(x)) = x^n. The lower triangular matrices for the coefficients of these two Appell sequences are a multiplicative inverse pair.
(End)
Sum_{k=0..n} (-1)^k * T(n,k) = A052845(n). - Alois P. Heinz, Feb 21 2022

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