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

0,5

Links

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

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)).

Example

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;

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[%]

Prog

(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
(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

Crossrefs

Cf. A000165, A007318, A060187, A142458.

Sequence in context: A323324 A181543 A141696 * A142470 A168523 A144439

Adjacent sequences: A178119 A178120 A178121 * A178123 A178124 A178125

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, May 20 2010

Extensions

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

Status

approved