A185164 -id:A185164 - cp's OEIS frontend

A008296 Triangle of Lehmer-Comtet numbers of the first kind.

1, 1, 1, -1, 3, 1, 2, -1, 6, 1, -6, 0, 5, 10, 1, 24, 4, -15, 25, 15, 1, -120, -28, 49, -35, 70, 21, 1, 720, 188, -196, 49, 0, 154, 28, 1, -5040, -1368, 944, 0, -231, 252, 294, 36, 1, 40320, 11016, -5340, -820, 1365, -987, 1050, 510, 45, 1, -362880, -98208, 34716, 9020, -7645, 3003, -1617, 2970, 825, 55, 1, 3628800

Offset: 1

N. J. A. Sloane

Triangle arising in the expansion of ((1+x)*log(1+x))^n.

Also the Bell transform of (-1)^(n-1)*(n-1)! if n>1 else 1 adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 16 2016

			Triangle begins:
   1;
   1,  1;
  -1,  3,   1;
   2, -1,   6,  1;
  -6,  0,   5, 10,  1;
  24,  4, -15, 25, 15, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139.

Alois P. Heinz, Rows n = 1..141, flattened
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y = x^x, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.
Tian-Xiao He and Yuanziyi Zhang, Centralizers of the Riordan Group, arXiv:2105.07262 [math.CO], 2021.
D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

Cf. A039621 (second kind), A354795 (variant), A185164, A005727 (row sums), A298511 (central).
Columns: A045406 (column 2), A347276 (column 3), A345651 (column 4).
Diagonals: A000142, A000217, A059302.
Cf. A176118.

for n from 1 to 20 do for k from 1 to n do
printf(`%d,`, add(binomial(l,k)*k^(l-k)*Stirling1(n,l), l=k..n)) od: od:
# second program:
A008296 := proc(n, k) option remember; if k=1 and n>1 then (-1)^n*(n-2)! elif n=k then 1 else (n-1)*procname(n-2, k-1) + (k-n+1)*procname(n-1, k) + procname(n-1, k-1) end if end proc:
seq(print(seq(A008296(n, k), k=1..n)), n=1..7); # Mélika Tebni, Aug 22 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n-2)!;
a[n_, n_] = 1; a[n_, k_] := a[n, k] = (n-1) a[n-2, k-1] + a[n-1, k-1] + (k-n+1) a[n-1,k]; Flatten[Table[a[n, k], {n, 1, 12}, {k, 1, n}]][[1 ;; 67]]
(* Jean-François Alcover, Apr 29 2011 *)

{T(n, k) = if( k<1 || k>n, 0, n! * polcoeff(((1 + x) * log(1 + x + x * O(x^n)))^k / k!, n))}; /* Michael Somos, Nov 15 2002 */

# uses[bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: (-1)^(n-1)*factorial(n-1) if n>1 else 1, 7) # Peter Luschny, Jan 16 2016

E.g.f. for a(n, k): (1/k!)[ (1+x)*log(1+x) ]^k. - Len Smiley
Left edge is (-1)*n!, for n >= 2. Right edge is all 1's.
a(n+1, k) = n*a(n-1, k-1) + a(n, k-1) + (k-n)*a(n, k).
a(n, k) = Sum_{m} binomial(m, k)*k^(m-k)*Stirling1(n, m).
From Peter Bala, Mar 14 2012: (Start)
E.g.f.: exp(t*(1 + x)*log(1 + x)) = Sum_{n>=0} R(n,t)*x^n/n! = 1 + t*x + (t+t^2)x^2/2! + (-t+3*t^2+t^3)x^3/3! + .... Cf. A185164. The row polynomials R(n,t) are of binomial type and satisfy the recurrence R(n+1,t) = (t-n)*R(n,t) + t*d/dt(R(n,t)) + n*t*R(n-1,t) with R(0,t) = 1 and R(1,t) = t. Inverse array is A039621.
(End)
Sum_{k=0..n} (-1)^k * a(n,k) = A176118(n). - Alois P. Heinz, Aug 25 2021

More terms from James Sellers, Jan 26 2001

Edited by N. J. A. Sloane at the suggestion of Andrew Robbins, Dec 11 2007

A039621 Triangle of Lehmer-Comtet numbers of 2nd kind.

Original entry on oeis.org

1, -1, 1, 4, -3, 1, -27, 19, -6, 1, 256, -175, 55, -10, 1, -3125, 2101, -660, 125, -15, 1, 46656, -31031, 9751, -1890, 245, -21, 1, -823543, 543607, -170898, 33621, -4550, 434, -28, 1, 16777216, -11012415, 3463615, -688506, 95781, -9702, 714, -36, 1

Offset: 1

Len Smiley

Also the Bell transform of (-n)^n adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 16 2016

			The triangle T(n, k) begins:
[1]       1;
[2]      -1,      1;
[3]       4,     -3,       1;
[4]     -27,     19,      -6,     1;
[5]     256,   -175,      55,   -10,     1;
[6]   -3125,   2101,    -660,   125,   -15,   1;
[7]   46656, -31031,    9751, -1890,   245, -21,   1;
[8] -823543, 543607, -170898, 33621, -4550, 434, -28, 1;

D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

A008296 (matrix inverse), A354794 (variant), A045531 (column |a(n, 2)|).

Cf. A185164.

R := proc(n, k, m) option remember;
   if k < 0  or n < 0 then 0 elif k = 0 then 1 else
   m*R(n, k-1, m) + R(n-1, k, m+1) fi end:
A039621 := (n, k) -> (-1)^(n-k)*R(k-1, n-k, n-k):
seq(seq(A039621(n, k), k = 1..n), n = 1..9); # Peter Luschny, Jun 10 2022 after Vladimir Kruchinin

a[1, 1] = 1; a[n_, k_] := 1/(k-1)! Sum[((-1)^(n-k-i)*Binomial[k-1, i]*(n-i-1)^(n-1)), {i, 0, k-1}];
Table[a[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* Jean-François Alcover, Jun 03 2019 *)

T(n,k,m):=if k<0 or n<0 then 0 else if k=0 then 1 else m*T(n,k-1,m)+T(n-1,k,m+1);
a(n,k):=if nVladimir Kruchinin, Mar 07 2020

tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(sum(i = 0, k-1,(-1)^(n-k-i)*binomial(k-1, i)*(n-i-1)^(n-1))/(k-1)!, ", ");); print(););} \\ Michel Marcus, Aug 28 2013

# uses[bell_matrix from A264428]
# Adds 1,0,0,0,... as column 0 at the left side of the triangle.
bell_matrix(lambda n: (-n)^n, 7) # Peter Luschny, Jan 16 2016

(k-1)!*a(n, k) = Sum_{i=0..k-1}((-1)^(n-k-i)*binomial(k-1, i)*(n-i-1)^(n-1)).

a(n,k) = (-1)^(n-k)*T(k,n-k,n-k), n>=k, where T(n,k,m)=m*T(n,m-1,k)+T(n-1,k,m+1), T(n,0,m)=1. - Vladimir Kruchinin, Mar 07 2020

A075856 Triangle formed from coefficients of the polynomials p(1)=x, p(n+1) = (n + x(n+1))p(n) + xx(d/dx)p(n).

Original entry on oeis.org

1, 1, 3, 2, 10, 15, 6, 40, 105, 105, 24, 196, 700, 1260, 945, 120, 1148, 5068, 12600, 17325, 10395, 720, 7848, 40740, 126280, 242550, 270270, 135135, 5040, 61416, 363660, 1332100, 3213210, 5045040, 4729725, 2027025

Offset: 1

F. Chapoton, Oct 15 2002

Constant terms of polynomials related to Ramanujan psi polynomials (see Zeng reference).

			Triangle begins
    1;
    1,    3;
    2,   10,   15;
    6,   40,  105,   105;
   24,  196,  700,  1260,   945;
  120, 1148, 5068, 12600, 17325, 10395;
  ...
p(1) = x, p(2) = 3*x^2 + x, p(3) = 15*x^3 + 10*x^2 + 2*x, etc. - _Michael Somos_, Mar 17 2011

P. Bala, Diagonals of triangles with generating function exp(t*F(x)).
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), #07.3.7.
Dominique Dumont and Armand Ramamonjisoa, Grammaire de Ramanujan et Arbres de Cayley, Electr. J. Combinatorics, Volume 3, Issue 2 (1996) R17 (see page 16).
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y = x^x, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.
M. Josuat-Vergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
S. Ramanujan, Notebook entry
P. W. Shor, Problem 78-6: A combinatorial identity, in Problems and Solutions column, SIAM Review; problem in 20, p. 394 (1978); solution in 21, pp. 258-260 (1979). [N. Sato, Feb 19 2010]
P. W. Shor, A = B (but not quite); 3-d array with multiple recurrences, MathOverflow, Nov 2010-Nov 2011.
J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, manuscript, 1996.
J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan J., 3(1999) 1, 45-54.

Cf. A000142, A000312, A001147, A054589. A185164, A209937.

See A239098 for another version.

p[1] = x; p[n_] := p[n] = (n - 1 + x*n)*p[n - 1] + x*x*D[p[n - 1], x]; Flatten[Rest[CoefficientList[#1, x]] & /@ Table[p[n], {n, 8}]] (* Jean-François Alcover, May 31 2011 *)

{T(n, k) = if( k<1 || nMichael Somos, Mar 17 2011 */

T(n, k) = (n-1) * T(n-1, k) + (n+k-1) * T(n-1, k-1). - Michael Somos, Mar 17 2011
G.f.: A(x, t) = Sum_{n>0} p[n] t^n / n! satisfies (dA / dt) * (x + t - 1) = x * (1 + A)^2 * (x * (1 + A) - 1). - Michael Somos, Mar 17 2011
T(n, 1) = (n-1)! = A000142(n-1). T(n, n) = A001147(n). Sum_{k>0} T(n, k) = n^n = A000312(n). Sum_{k>0} T(n, k) x^k = p[n].
From Peter Bala, Mar 14 2012: (Start)
This triangle is A185164 read by diagonals.
Let F(x) = x + (1-x)*log(1-x). The e.g.f. is given by the compositional inverse
(x - t*F(x))^(-1) = x + t*x^2/2! + (t + 3*t^2)x^3/3! + (2*t + 10*t^2 + 15*t^3)*x^4/4! + ....
Let f(x) = 1/log(1+x) and define inductively D^(n+1)(f(x)) = f(x)*(d/dx)(D^n(f(x))) with D^(0)f(x) = f(x). Then D^(n)f = (-1)^n*Sum_{k = 1..n} T(n,k)*f^(n-k)/((1+x)^n*f^(2n+1)).
(End)

A203852 Expansion of e.g.f. exp( Integral -log(1-x) dx ).

Original entry on oeis.org

1, 0, 1, 1, 5, 16, 79, 421, 2673, 19216, 156021, 1411873, 14117773, 154730720, 1845959179, 23826445501, 330951133537, 4923574598112, 78123812086441, 1317174439409409, 23517962293307701, 443340968936640496, 8799729204814165223, 183448995762912568885

Offset: 0

Paul D. Hanna, Jan 29 2012

nonn

Row sums of A185164. - Peter Bala, Mar 14 2012

			E.g.f.: A(x) = 1 + x^2/2! + x^3/3! + 5*x^4/4! + 16*x^5/5! + 79*x^6/6! +...
where: log(A(x)) = x^2/2 + x^3/6 + x^4/12 + x^5/20 + x^6/30 + x^7/42 +...

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A185164.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)*(1-x)^(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 17 2018

CoefficientList[Series[Exp[x]*(1-x)^(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 27 2013 *)

a(n):=if n=0 then 1 else sum(binomial(n-1,i)*(i-1)!*a(n-i-1),i,1,n-1); /* Vladimir Kruchinin, Feb 23 2015 */

{a(n)=n!*polcoeff(exp(-intformal(log(1-x +x*O(x^n)))), n)}

x='x+O('x^30); Vec(serlaplace(exp(x)*(1-x)^(1-x))) \\ G. C. Greubel, Jul 17 2018

E.g.f.: exp( Sum_{n>=2} x^n/(n*(n-1)) ).
E.g.f.: exp(x)*(1-x)^(1-x). - Vaclav Kotesovec, Dec 27 2013
a(n) ~ (n-1)! * (exp(1)/n + (2*log(n)+2*gamma)/n^2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Dec 27 2013
a(n) = sum(i=1..n-1, binomial(n-1,i)*(i-1)!*a(n-i-1),i,1,n-1), a(0)=1. - Vladimir Kruchinin, Feb 23 2015

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A008296 Triangle of Lehmer-Comtet numbers of the first kind.

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A039621 Triangle of Lehmer-Comtet numbers of 2nd kind.

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A075856 Triangle formed from coefficients of the polynomials p(1)=x, p(n+1) = (n + x*(n+1))*p(n) + x*x*(d/dx)p(n).

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A203852 Expansion of e.g.f. exp( Integral -log(1-x) dx ).

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A075856 Triangle formed from coefficients of the polynomials p(1)=x, p(n+1) = (n + x(n+1))p(n) + xx(d/dx)p(n).