A152818 - OEIS

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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

(list; table; graph; refs; listen; history; text; internal format)

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, 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) >;