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

%I #8 Mar 18 2022 03:08:55

%S 1,1,1,1,8,1,1,27,27,1,1,82,240,82,1,1,245,1700,1700,245,1,1,732,

%T 10571,23586,10571,732,1,1,2191,60697,259791,259791,60697,2191,1,1,

%U 6566,331666,2485398,4675152,2485398,331666,6566,1,1,19689,1756410,21708138,69413544,69413544,21708138,1756410,19689,1

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

%H G. C. Greubel, <a href="/A178122/b178122.txt">Rows n = 0..50 of the triangle, flattened</a>

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

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

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

%e Rows n>=0 and columns 0<=m<=n start as:

%e 1;

%e 1, 1;

%e 1, 8, 1;

%e 1, 27, 27, 1;

%e 1, 82, 240, 82, 1;

%e 1, 245, 1700, 1700, 245, 1;

%e 1, 732, 10571, 23586, 10571, 732, 1;

%e 1, 2191, 60697, 259791, 259791, 60697, 2191, 1;

%e 1, 6566, 331666, 2485398, 4675152, 2485398, 331666, 6566, 1;

%e 1, 19689, 1756410, 21708138, 69413544, 69413544, 21708138, 1756410, 19689, 1;

%t p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];

%t f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];

%t t[n_, m_] := f[n, m] + 2*Binomial[n, m] - 2 ;

%t Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];

%t Flatten[%]

%o (Magma)

%o A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;

%o A178122:= func< n,k | A060187(n+1, k+1) + 2*Binomial(n, k) - 2 >;

%o [A178122(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 18 2022

%o (Sage)

%o def A060187(n,k): return sum( (-1)^(k-j)*binomial(n, k-j)*(2*j-1)^(n-1) for j in (1..k) )

%o def A178122(n,k): return A060187(n+1, k+1) + 2*binomial(n, k) - 2

%o flatten([[A178122(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 18 2022

%Y Cf. A000165, A007318, A060187, A142458.

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_, May 20 2010

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