A176082 Triangle, read by rows, T(n,k) = f(n,k) - f(n,0) + 1, where f(n,k) = Sum_{j=0..k} (n+k)!/((n-j)!*(k-j)!*j!) + Sum_{j=0..n-k} (2*n-k)!/((n-j)!*(n - k-j)!*j!).

1, 1, 1, 1, -24, 1, 1, -534, -534, 1, 1, -12050, -14000, -12050, 1, 1, -326430, -381325, -381325, -326430, 1, 1, -10442537, -12093494, -12275676, -12093494, -10442537, 1, 1, -384746012, -441975660, -448634214, -448634214, -441975660, -384746012, 1

Offset

0,5

Comments

Row sums are: {1, 2, -22, -1066, -38098, -1415508, -57347736, -2550711770, -124197093898, -6585168718564, -378067505752484, ...}.

Links

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

Formula

T(n,k) = f(n,k) - f(n,0) + 1, where f(n,k) = Sum_{j=0..k} (n+k)!/((n-j)!*(k-j)! *j!) + Sum_{j=0..n-k} (2*n-k)!/((n-j)!*(n-k-j)!*j!).

Example

Triangle begins as:
  1;
  1,         1;
  1,       -24,         1;
  1,      -534,      -534,         1;
  1,    -12050,    -14000,    -12050,         1;
  1,   -326430,   -381325,   -381325,   -326430,         1;
  1, -10442537, -12093494, -12275676, -12093494, -10442537, 1;

Maple

b:=binomial; f(n, k):=b(n+k, n)*add(j!*b(n, j)*b(k, j), j=0..k) + b(2*n-k, n)*add( j!*b(n, j)*b(n-k, j), j=0..n-k); seq(seq(f(n, k)-f(n, 0)+1, k=0..n), n=0..10); # G. C. Greubel, Nov 27 2019

Mathematica

f[n_, k_]:= Sum[(n+k)!/((n-j)!*(k-j)!*j!), {j, 0, k}] + Sum[(n-k)!/((n-j)!*(n-k- j)!*j!), {j, 0, n-k}]; Table[f[n, k] -f[n, 0] +1, {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(PARI) b=binomial; f(n, k) = b(n+k, n)*sum(j=0, k, j!*b(n, j)*b(k, j)) + b(2*n-k, n)* sum(j=0, n-k, j!*b(n, j)*b(n-k, j));
T(n, k) = f(n, k) - f(n, 0) + 1; \\ G. C. Greubel, Nov 27 2019
(Magma)
function f(n, k)
    B:=Binomial;
  return B(n+k, n)*(&+[Factorial(j)*B(n, j)*B(k, j): j in [0..k]]) + B(2*n-k, n)* (&+[Factorial(j)*B(n, j)*B(n-k, j): j in [0..n-k]]); end function;
[f(n, k) -f(n, 0) +1: k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 27 2019
(Sage)
def f(n, k):
    b=binomial;
    return b(n+k, n)*sum(factorial(j)*b(n, j)*b(k, j) for j in (0..k)) + b(2*n-k, n)*sum(factorial(j)*b(n, j)*b(n-k, j) for j in (0..n-k))
[[f(n, k) -f(n, 0) +1 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 27 2019
(GAP)
B:=Binomial;;
f:= function(n, k) return B(n+k, n)*Sum([0..k], j-> Factorial(j)*B(n, j)*B(k, j)) + B(2*n-k, n)*Sum([0..n-k], j-> Factorial(j)*B(n, j)*B(n-k, j));  end;
Flat(List([0..10], n-> List([0..n], k-> f(n, k)-f(n, 0)+1 ))); # G. C. Greubel, Nov 27 2019

Crossrefs

Cf. A176081.

Sequence in context: A040578 A040577 A040576 * A022187 A040586 A040587

Adjacent sequences: A176079 A176080 A176081 * A176083 A176084 A176085

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 08 2010

Status

approved