A176490 - OEIS

%I #9 Jun 17 2015 04:01:54

%S 1,1,1,1,9,1,1,33,33,1,1,101,295,101,1,1,293,1983,1983,293,1,1,841,

%T 11733,25963,11733,841,1,1,2425,64949,275341,275341,64949,2425,1,1,

%U 7053,346219,2573521,4831203,2573521,346219,7053,1,1,20685,1804179,22163163

%N Triangle T(n,k) = A008292(n+1,k+1) + A060187(n+1,k+1)- 1 read along rows 0<=k<=n.

%C Row sums are: 1, 2, 11, 68, 499, 4554, 51113, 685432, 10684791, 189423350, 3755807989,....

%C Conjecture on the row sums s(n): 859*(n+1)*s(n) +(-2577*n^2-15955*n+33324)*s(n-1) +(1718*n^3+39275*n^2-102106*n-16383)*s(n-2) +(-25038*n^3+35127*n^2+252701*n-453082)*s(n-3) +(n-3)*(57834*n^2-211893*n+212386)*s(n-4) -2*(17257*n-29530)*(n-3)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jun 16 2015

%e 1;

%e 1, 1;

%e 1, 9, 1;

%e 1, 33, 33, 1;

%e 1, 101, 295, 101, 1;

%e 1, 293, 1983, 1983, 293, 1;

%e 1, 841, 11733, 25963, 11733, 841, 1;

%e 1, 2425, 64949, 275341, 275341, 64949, 2425, 1;

%e 1, 7053, 346219, 2573521, 4831203, 2573521, 346219, 7053, 1;

%e 1, 20685, 1804179, 22163163, 70723647, 70723647, 22163163, 1804179, 20685, 1;

%e 1, 61073, 9268777, 180504391, 916661395, 1542816715, 916661395, 180504391, 9268777, 61073, 1;

%p A176490 := proc(n,k)

%p     A008292(n+1,k+1)+A060187(n+1,k+1)-1 ;

%p end proc: # _R. J. Mathar_, Jun 16 2015

%t (*A060187*)

%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 << DiscreteMath`Combinatorica`;

%t t[n_, m_, 0] := Binomial[n, m];

%t t[n_, m_, 1] := Eulerian[1 + n, m];

%t t[n_, m_, 2] := f[n, m];

%t t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;

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

%Y Cf. A007318, A008292, A060187, A176487.

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_, Apr 19 2010