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

0,4

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.

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Index entries for sequences related to Laguerre polynomials

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)

Example

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 + ...

Mathematica

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

Prog

(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
(Magma) [Factorial(n)*Evaluate(LaguerrePolynomial(n-k, k-1), -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021
(PARI) T(n, k) = n! * pollaguerre(n-k, k-1, -1); \\ Michel Marcus, Feb 23 2021

Crossrefs

Cf. A000262, A052852, A052897, A059110.

Row sums give A059115. Alternating row sums give A288268.

Sequence in context: A366896 A072565 A159672 * A246322 A166074 A225475

Adjacent sequences: A059111 A059112 A059113 * A059115 A059116 A059117

Keyword

easy,nonn,tabl

Author

Vladeta Jovovic, Jan 04 2001

Status

approved