A178122 Triangle T(n,m) = A060187(n+1,m+1) + 2*binomial(n,m) - 2, read by rows.

1, 1, 1, 1, 8, 1, 1, 27, 27, 1, 1, 82, 240, 82, 1, 1, 245, 1700, 1700, 245, 1, 1, 732, 10571, 23586, 10571, 732, 1, 1, 2191, 60697, 259791, 259791, 60697, 2191, 1, 1, 6566, 331666, 2485398, 4675152, 2485398, 331666, 6566, 1, 1, 19689, 1756410, 21708138, 69413544, 69413544, 21708138, 1756410, 19689, 1

Offset: 0

Author

Roger L. Bagula, May 20 2010

Keywords

• nonn
• tabl
• easy

Examples

			Rows n>=0 and columns 0<=m<=n start as:
  1;
  1,     1;
  1,     8,       1;
  1,    27,      27,        1;
  1,    82,     240,       82,        1;
  1,   245,    1700,     1700,      245,        1;
  1,   732,   10571,    23586,    10571,      732,        1;
  1,  2191,   60697,   259791,   259791,    60697,     2191,       1;
  1,  6566,  331666,  2485398,  4675152,  2485398,   331666,    6566,     1;
  1, 19689, 1756410, 21708138, 69413544, 69413544, 21708138, 1756410, 19689, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000165, A007318, A060187, A142458.

Programs

Magma

A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;
A178122:= func< n,k | A060187(n+1, k+1) + 2*Binomial(n, k) - 2 >;
[A178122(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2022

Mathematica

p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
t[n_, m_] := f[n, m] + 2*Binomial[n, m] - 2 ;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Sage

def A060187(n,k): return sum( (-1)^(k-j)*binomial(n, k-j)*(2*j-1)^(n-1) for j in (1..k) )
def A178122(n,k): return A060187(n+1, k+1) + 2*binomial(n, k) - 2
flatten([[A178122(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

Formula

T(n, m) = A060187(n+1,m+1) + 2*A007318(n,m) - 2.

T(n, m) = T(n, n-m).

Sum_{k=0..n} T(n, k) = A000165(n) + 2*(2^n -(n+1)).

Extensions

Indices in definition corrected, row sum formula added by the Assoc. Eds. of the OEIS - Aug 20 2010