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

%I A178122 #8 Mar 18 2022 03:08:55
%S A178122 1,1,1,1,8,1,1,27,27,1,1,82,240,82,1,1,245,1700,1700,245,1,1,732,
%T A178122 10571,23586,10571,732,1,1,2191,60697,259791,259791,60697,2191,1,1,
%U A178122 6566,331666,2485398,4675152,2485398,331666,6566,1,1,19689,1756410,21708138,69413544,69413544,21708138,1756410,19689,1
%N A178122 Triangle T(n,m) = A060187(n+1,m+1) + 2*binomial(n,m) - 2, read by rows.
%H A178122 G. C. Greubel, <a href="/A178122/b178122.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A178122 T(n, m) = A060187(n+1,m+1) + 2*A007318(n,m) - 2.
%F A178122 T(n, m) = T(n, n-m).
%F A178122 Sum_{k=0..n} T(n, k) = A000165(n) + 2*(2^n -(n+1)).
%e A178122 Rows n>=0 and columns 0<=m<=n start as:
%e A178122   1;
%e A178122   1,     1;
%e A178122   1,     8,       1;
%e A178122   1,    27,      27,        1;
%e A178122   1,    82,     240,       82,        1;
%e A178122   1,   245,    1700,     1700,      245,        1;
%e A178122   1,   732,   10571,    23586,    10571,      732,        1;
%e A178122   1,  2191,   60697,   259791,   259791,    60697,     2191,       1;
%e A178122   1,  6566,  331666,  2485398,  4675152,  2485398,   331666,    6566,     1;
%e A178122   1, 19689, 1756410, 21708138, 69413544, 69413544, 21708138, 1756410, 19689, 1;
%t A178122 p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
%t A178122 f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
%t A178122 t[n_, m_] := f[n, m] + 2*Binomial[n, m] - 2 ;
%t A178122 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A178122 Flatten[%]
%o A178122 (Magma)
%o A178122 A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;
%o A178122 A178122:= func< n,k | A060187(n+1, k+1) + 2*Binomial(n, k) - 2 >;
%o A178122 [A178122(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 18 2022
%o A178122 (Sage)
%o A178122 def A060187(n,k): return sum( (-1)^(k-j)*binomial(n, k-j)*(2*j-1)^(n-1) for j in (1..k) )
%o A178122 def A178122(n,k): return A060187(n+1, k+1) + 2*binomial(n, k) - 2
%o A178122 flatten([[A178122(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 18 2022
%Y A178122 Cf. A000165, A007318, A060187, A142458.
%K A178122 nonn,tabl,easy
%O A178122 0,5
%A A178122 _Roger L. Bagula_, May 20 2010
%E A178122 Indices in definition corrected, row sum formula added by the Assoc. Eds. of the OEIS - Aug 20 2010