A059114 Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.

1, 1, 1, 3, 4, 2, 13, 21, 18, 6, 73, 136, 156, 96, 24, 501, 1045, 1460, 1260, 600, 120, 4051, 9276, 15030, 16320, 11160, 4320, 720, 37633, 93289, 170142, 219450, 192360, 108360, 35280, 5040, 394353, 1047376, 2107448, 3116736, 3294480, 2405760, 1149120, 322560, 40320

Offset: 0

Author

Vladeta Jovovic, Jan 04 2001

Keywords

• easy
• nonn
• tabl

Comments

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.

Examples

			Triangle begins as:
    1;
    1,    1;
    3,    4,    2;
   13,   21,   18,    6;
   73,  136,  156,   96,  24;
  501, 1045, 1460, 1260, 600, 120;
  ...;
E.g.f. for T(n, 2) = (x/(1-x))^2*e^(x/(x-1)) = x^2 + 3*x^3 + 13/2*x^4 + 73/6*x^5 + 167/8*x^6 + 4051/120*x^7 + ...
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A000262, A052852, A052897, A059110.

Row sums give A059115. Alternating row sums give A288268.

Programs

Magma

[Factorial(n)*Evaluate(LaguerrePolynomial(n-k, k-1), -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

Mathematica

Table[n!*LaguerreL[n-k, k-1, -1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

PARI

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

Sage

flatten([[factorial(n)*gen_laguerre(n-k, k-1, -1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

Formula

E.g.f. for T(n, k) = (x/(1-x))^k * exp(x/(x-1)).

T(n, k)= Sum_{i=0..n} L'(n,i) * ( Product_{j=1..k} (i-j+1) ).

T(n, 0) = A000262(n).

T(n, 1) = A052852(n).

From G. C. Greubel, Feb 23 2021: (Start)

T(n, k) = n! * k! * Sum_{j=0..n} binomial(j, k)*binomial(n-1, j-1)/j!.

T(n, k) = n! * Laguerre(n-k, k-1, -1).

T(n, k) = n!*binomial(n-1, k-1)*Hypergeometric1F1([k-n], [k], -1) with T(n, 0) = Hypergeometric2F0([1-n, -n], [], 1). (End)