A059099 -id:A059099 - cp's OEIS frontend

A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).

1, 2, 9, 58, 485, 4986, 60877, 861554, 13878153, 250854130, 5030058161, 110837000682, 2662669300909, 69270266115818, 1940260799150325, 58220372514830626, 1863293173842259217, 63356877145370671074

Offset: 0

Vladeta Jovovic, Jan 06 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.

			(1-x)/(1-2*x)*exp(x/(1-x)) = 1 + 2*x + 9/2*x^2 + 29/3*x^3 + 485/24*x^4 + 831/20*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for sequences related to Laguerre polynomials

Cf. A001861, A049020, A052897, A059099, A059110.

Row sums of A059114.

[Factorial(n)*(&+[Evaluate(LaguerrePolynomial(n-k, k-1), -1) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 23 2021

s := series((1-x)/(1-2*x)*exp(x/(1-x)), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

With[{nn=20},CoefficientList[Series[(1-x)/(1-2x) Exp[x/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2020 *)
Table[n!*Sum[LaguerreL[n-k, k-1, -1], {k,0,n}], {n,0,30}] (* G. C. Greubel, Feb 23 2021 *)

{a(n)=if(n<0, 0, n!*polcoeff( (1-x)/(1-2*x)*exp(x/(1-x)+x*O(x^n)), n))} /* Michael Somos, Aug 03 2006 */

{a(n)=local(A); if(n<0,0, n++; A=vector(n); A[n]=1; for(k=1,n-1, A[n-k]=1; if(k>1, A[n-k+1]=A[n-k+2]); for(i=n-k+1,n, A[i]=A[i-1]+k*A[i])); A[n])} /* Michael Somos, Aug 03 2006 */

a(n) = n!*sum(k=0, n, pollaguerre(n-k, k-1, -1)); \\ Michel Marcus, Feb 23 2021

[factorial(n)*sum( gen_laguerre(n-k, k-1, -1) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Feb 23 2021

Sum_{m=0..n} Sum_{i=0..n} L'(n, i)*Product_{j=1..m} (i-j+1).
Given g.f. A(x), then g.f. A000522 = A(x/(1+x)). - Michael Somos, Aug 03 2006
a(n) = n!*Sum_{k=0..n} LaguerreL(n-k, k-1, -1). - G. C. Greubel, Feb 23 2021
a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n-1). - Vaclav Kotesovec, Feb 23 2021

Definition clarified by Harvey P. Dale, Jul 18 2020

A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 5, 10, 12, 6, 15, 37, 62, 60, 24, 52, 151, 320, 450, 360, 120, 203, 674, 1712, 3120, 3720, 2520, 720, 877, 3263, 9604, 21336, 33600, 34440, 20160, 5040, 4140, 17007, 56674, 147756, 287784, 394800, 352800, 181440, 40320, 21147, 94828

Offset: 0

Vladeta Jovovic, Jan 02 2001

The transpose of this lower unitriangular array is the U factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20L%20factor%20is%20A049020%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The L factor is A049020 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

			Triangle begins:
  [0] [ 1]
  [1] [ 1,    1]
  [2] [ 2,    3,    2]
  [3] [ 5,   10,   12,    6]
  [4] [15,   37,   62,   60,   24]
  [5] [52,  151,  320,  450,  360,  120]
  [6] [203, 674, 1712, 3120, 3720, 2520, 720]
  ...;
E.g.f. for T(n, 2) = (exp(x)-1)^2*(exp(exp(x)-1)) = x^2 + 2*x^3 + 31/12*x^4 + 8/3*x^5 + 107/45*x^6 + 343/180*x^7 + 28337/20160*x^8 + 349/360*x^9 + ...;
E.g.f. for T(n, 3) = (exp(x)-1)^3*(exp(exp(x)-1)) = x^3 + 5/2*x^4 + 15/4*x^5 + 13/3*x^6 + 127/30*x^7 + 1759/480*x^8 + 34961/12096*x^9 + ...

Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

Cf. A000110(n) = T(n,0), A005493(n) = T(n,1), A059099 (row sums).

Cf. A049020, A001861, A046716.

T := proc(n, k) option remember; `if`(k < 0 or k > n, 0,
      `if`(n = 0, 1, k*T(n-1, k-1) + (k+1)*T(n-1, k) + T(n-1, k+1)))
    end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..15); # Peter Bala, Oct 15 2023

E.g.f. for T(n, k): (exp(x)-1)^k*(exp(exp(x)-1)).
n-th row is M^n*[1,0,0,0,...], where M is a tridiagonal matrix with all 1's in the superdiagonal, (1,2,3,...) in the main and subdiagonals; and the rest zeros. - Gary W. Adamson, Jun 23 2011
T(n, k) = k!*A049020(n, k). - R. J. Mathar, May 17 2016
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j). - Peter Luschny, Dec 06 2023

A227343 Matrix inverse of triangle A227342.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 13, 13, 6, 1, 75, 75, 37, 10, 1, 541, 541, 270, 85, 15, 1, 4683, 4683, 2341, 770, 170, 21, 1, 47293, 47293, 23646, 7861, 1890, 308, 28, 1, 545835, 545835, 272917, 90930, 22491, 4158, 518, 36, 1, 7087261, 7087261, 3543630, 1181125, 294525, 57351, 8400, 822, 45, 1

Offset: 0

Peter Bala, Jul 11 2013

The e.g.f. has the form A(t)*exp(x*B(t)), where A(t) = 1/(2 - exp(t)) and B(t) = exp(t) - 1. Thus the row polynomials of this triangle form a Sheffer sequence for the pair (1 - t, log(1 + t)) (see Roman, p.17).

Let x_(k) := x*(x-1)*...*(x-k+1) denote the k-th falling factorial polynomial. Define a sequence x_[n] of basis polynomials for the polynomial algebra C[x] by setting x_[0] = 1, and setting x_[n] = x_(n-1)*(x - 2*n + 1) for n >= 1. The sequence begins [1, x-1, x*(x-3), x*(x-1)*(x-5), x*(x-1)*(x-2)*(x-7), ...]. Then this is the triangle of connection constants for expressing the monomial polynomials x^n as a linear combination of the basis x_[k], that is, x^n = sum {k = 0..n} T(n,k)*x_[k]. An example is given below.

			Triangle begins
n\k|   0    1    2    3    4    5
= = = = = = = = = = = = = = = = =
0 |   1
1 |   1    1
2 |   3    3    1
3 |  13   13    6    1
4 |  75   75   37   10    1
5 | 541  541  270   85   15    1
...
Connection constants. Row 4 = [75,75,37,10,1]: Thus
75 + 75*(x - 1) + 37*x*(x - 3) + 10*x*(x - 1)*(x - 5)+ x*(x - 1)*(x - 2)*(x - 7) = x^4.

S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press Inc., New York, 1984. Reprinted by Dover in 2005.

Eric Weisstein's World of Mathematics, Sheffer Sequence

A000670 (columns 1 and 2), A048993, A059099 (row sums), A105794, A227342 (matrix inverse).

T[n_, k_] := n!/k! SeriesCoefficient[Series[1/(2 - Exp[t]) (Exp[t] - 1)^k, {t, 0, n}], n]
Flatten[Table[T[n, k], {n, 0, 12}, {k, 0, n}]]
U[n_, k_] := n!/k! SeriesCoefficient[Series[1/(1 - t^2) (t/Log[1 + t])^(n + 1), {t, 0, n - k}], n - k]
Flatten[Table[U[n, k], {n, 0, 8}, {k, 0, n}]] (* Emanuele Munarini, Dec 21 2016 *)

E.g.f.: 1/(2 - exp(t))*exp(x*(exp(t) - 1)) = 1 + (1 + x)*t + (3 + 3*x + x^2)*t^2/2! + (13 + 13*x + 6*x^2 + x^3)*t^3/3! + ....
Recurrence equation: T(n,0) = A000670(n), and for k >= 1, T(n,k) = 1/k*sum {i = 1..n} binomial(n,i)*T(n-i,k-1).
The row polynomials R(n,x) satisfy the Sheffer identity R(n,x + y) = sum {k = 0..n} binomial(n,k)*Bell(k,y)*R(n-k,x), where Bell(k,y) is the Bell or exponential polynomial (row polynomials of A048993).
The row polynomials also satisfy d/dx(R(n,x)) = sum {k = 0..n-1} binomial(n,k)*R(k,x).
Row sums A059099. Column 1 and column 2 = A000670. 1 + 2*column 3 = A000670 (apart from the first two terms).
From Emanuele Munarini, Dec 21 2016: (Start)
T(n,k) = (n!/k!)*[t^n](exp(t)-1)^k/(2-exp(t)).
T(n,k) = (n!/k!)*[t^(n-k)](t/log(1+t))^(n+1)/(1-t^2). (End)

A331797 E.g.f.: (exp(x) - 1) * exp(exp(x) - 1) / (2 - exp(x)).

Original entry on oeis.org

0, 1, 5, 28, 183, 1401, 12466, 127443, 1478581, 19239274, 277797577, 4409962349, 76355817104, 1432117088325, 28925947345561, 625973017346996, 14449435509751843, 354384392492622789, 9202836581079864186, 252260861877820739167, 7278710020682729662089

Offset: 0

Ilya Gutkovskiy, Jan 26 2020

nonn

Stirling transform of A007526.

Cf. A000110, A000670, A007526, A059099, A331796, A331798.

nmax = 20; CoefficientList[Series[(Exp[x] - 1) Exp[Exp[x] - 1]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
A007526[n_] := n! Sum[1/k!, {k, 0, n - 1}]; a[n_] := Sum[StirlingS2[n, k] A007526[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, -k, 0] BellB[n - k], {k, 1, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A007526(k).
a(n) = Sum_{k=1..n} binomial(n,k) * A000670(k) * A000110(n-k).
a(n) ~ n! * exp(1) / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jan 26 2020

A306038 Expansion of e.g.f. (1 + x)/(1 - log(1 + x)).

Original entry on oeis.org

1, 2, 3, 5, 12, 34, 122, 482, 2328, 11640, 71952, 424368, 3312240, 21357504, 217045488, 1351338864, 19990187520, 89379824256, 2631270916224, 892036259712, 507945420198144, -3068802187635456, 142961233091051520, -1849617314640322560, 55640352746480440320

Offset: 0

Ilya Gutkovskiy, Jun 17 2018

sign

			(1 + x)/(1 - log(1 + x)) = 1 + 2*x + 3*x^2/2! + 5*x^3/3! + 12*x^4/4! + 34*x^5/5! + 122*x^6/6! + ...

Robert Israel, Table of n, a(n) for n = 0..452
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000522, A006252, A059099, A089064.

S:= series((1+x)/(1-log(1+x)),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Jun 19 2018

nmax = 24; CoefficientList[Series[(1 + x)/(1 - Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
FullSimplify[Table[Sum[StirlingS1[n, k] E Gamma[1 + k, 1], {k, 0, n}], {n, 0, 24}]]

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000522(k).

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A059115 Expansion of e.g.f.: ((1-x)/(1-2*x))*exp(x/(1-x)).

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A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

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A227343 Matrix inverse of triangle A227342.

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A331797 E.g.f.: (exp(x) - 1) * exp(exp(x) - 1) / (2 - exp(x)).

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A306038 Expansion of e.g.f. (1 + x)/(1 - log(1 + x)).

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A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).