A152818 Array read by antidiagonals: A(n,k) = (k+1)^n*(n+k)!/n!.

1, 1, 1, 1, 4, 2, 1, 12, 18, 6, 1, 32, 108, 96, 24, 1, 80, 540, 960, 600, 120, 1, 192, 2430, 7680, 9000, 4320, 720, 1, 448, 10206, 53760, 105000, 90720, 35280, 5040, 1, 1024, 40824, 344064, 1050000, 1451520, 987840, 322560, 40320

Offset

0,5

Comments

A009998/A119502 gives triangle of unreduced coefficients of polynomials defined by A152650/A152656. a(n) gives numerators with denominators n! for each row.

Row 0 is A000142. Row 1 is formed from positive members of A001563. Row 2 is A055533. Column 0 is A000012. Column 1 is formed from positive members of A001787. Column 2 is A006043. Column 3 is A006044. - Omar E. Pol, Jan 06 2009

Links

G. C. Greubel, Antidiagonals n = 0..50, flattened

F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. See page 422.

F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)

F. A. Haight, Letter to N. J. A. Sloane, n.d.

Formula

E.g.f. for array as a triangle: exp(x)/(1-t*x*exp(x)) = 1+(1+t)*x+(1+4*t+2*t^2)*x^2/2! + (1+12*t+18*t^2+6*t^3)*x^3/3! + .... E.g.f. is int {z = 0..inf} exp(-z)*F(x,t*z), (x and t chosen sufficiently small for the integral to converge), where F(x,t) = exp(x*(1+t*exp(x))) is the e.g.f. for A154372. - Peter Bala, Oct 09 2011

From Peter Bala, Oct 09 2011: (Start)

From the e.g.f., the row polynomials R(n,t) satisfy the recursion R(n,t) = 1 + t*sum {k = 0..n-1} n!/(k!*(n-k-1)!)*R(n-k-1,t). The polynomials 1/n!*R(n,x) are the polynomials P(n,x) of A152650.

Sum_{k=0..n} T(n, k) = A072597(n) (antidiagonal sums). (End)

From G. C. Greubel, Apr 10 2023: (Start)

T(n, k) = (k+1)^(n-k) * k! * binomial(n, k) (antidiagonal triangle).

Sum_{k=0..n} (-1)^k*T(n, k) = A089148(n). (End)

Example

From Omar E. Pol, Jan 06 2009: (Start)
Array begins:
  1,    1,      2,        6,         24,          120, ...
  1,    4,     18,       96,        600,         4320, ...
  1,   12,    108,      960,       9000,        90720, ...
  1,   32,    540,     7680,     105000,      1451520, ...
  1,   80,   2430,    53760,    1050000,     19595520, ...
  1,  192,  10206,   344064,    9450000,    235146240, ...
  1,  448,  40824,  2064384,   78750000,   2586608640, ...
  1, 1024, 157464, 11796480,  618750000,  26605117440, ...
  1, 2304, 590490, 64880640, 4640625000, 259399895040, ... (End)
Antidiagonal triangle:
  1;
  1,   1;
  1,   4,     2;
  1,  12,    18,     6;
  1,  32,   108,    96,     24;
  1,  80,   540,   960,    600,   120;
  1, 192,  2430,  7680,   9000,  4320,   720;
  1, 448, 10206, 53760, 105000, 90720, 35280, 5040;

Mathematica

len= 45; m= 1 + Ceiling[Sqrt[len]]; Sort[Flatten[#, 1] &[MapIndexed[ {(2 +#2[[1]]^2 +(#2[[2]] -1)*#2[[2]] +#2[[1]]*(2*#2[[2]] -3))/ 2, #1}&, Table[(k+1)^n*(n+k)!/n!, {n, 0, m}, {k, 0, m}], {2}]]][[All, 2]][[1 ;; len]] (* From Jean-François Alcover, May 27 2011 *)
T[n_, k_]:= (k+1)^(n-k)*k!*Binomial[n, k];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 10 2023 *)

Prog

(Sage)
def A152818_row(n):
    R.<x> = ZZ[]
    P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))
    return P.coefficients()
for n in (0..12): print(A152818_row(n))  # Peter Luschny, May 03 2013
(PARI) A(n, k) = (k+1)^n*(n+k)!/n! \\ Charles R Greathouse IV, Sep 10 2016
(Magma)
A152818:= func< n, k | (k+1)^(n-k)*Factorial(k)*Binomial(n, k) >;
[A152818(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2023

Crossrefs

Cf. A000012, A000142, A001563, A001787, A006043, A006044, A009998.

Cf. A055533, A072597 (antidiagonal sums), A089148, A119502, A152650.

Cf. A152656, A154372.

Sequence in context: A328647 A183158 A174005 * A302235 A242861 A109244

Adjacent sequences: A152815 A152816 A152817 * A152819 A152820 A152821

Keyword

nonn,tabl

Author

Paul Curtz, Dec 13 2008

Extensions

Better definition, extended and edited by Omar E. Pol and N. J. A. Sloane, Jan 05 2009

Status

approved