A155493 - OEIS

%I #7 Apr 01 2022 18:23:43

%S 1,1,1,1,15,1,1,118,118,1,1,770,3540,770,1,1,4671,67810,67810,4671,1,

%T 1,27321,1039689,3085355,1039689,27321,1,1,156220,14006244,99524810,

%U 99524810,14006244,156220,1,1,878868,173788752,2602528824,6090918372,2602528824,173788752,878868,1

%N Triangle T(n, k) = binomial(n+1, k)*A142459(n+1, k+1)/(k+1), read by rows.

%H G. C. Greubel, <a href="/A155493/b155493.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = binomial(n+1, k)*t(n, k, m)/(k+1), where t(n,k,m) = (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m), t(n,1,m) = t(n,n,m) = 1, and m = 4.

%F From _G. C. Greubel_, Apr 01 2022: (Start)

%F T(n, k) = binomial(n+1, k)*A142459(n+1, k+1)/(k+1).

%F T(n, n-k) = T(n, k). (End)

%e Triangle begins as:

%e   1;

%e   1,      1;

%e   1,     15,         1;

%e   1,    118,       118,          1;

%e   1,    770,      3540,        770,          1;

%e   1,   4671,     67810,      67810,       4671,          1;

%e   1,  27321,   1039689,    3085355,    1039689,      27321,         1;

%e   1, 156220,  14006244,   99524810,   99524810,   14006244,    156220,      1;

%e   1, 878868, 173788752, 2602528824, 6090918372, 2602528824, 173788752, 878868, 1;

%t t[n_, k_, m_]:= t[n,k,m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k -(m -1))*t[n-1,k,m]];

%t T[n_, k_, m_]:= Binomial[n+1,k]*t[n+1,k+1,m]/(k+1);

%t Table[T[n,k,4], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 01 2022 *)

%o (Sage)

%o @CachedFunction

%o def t(n,k,m):

%o     if (k==1 or k==n): return 1

%o     else: return (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m)

%o def T(n,k,m): return binomial(n+1,k)*t(n+1,k+1,m)/(k+1)

%o flatten([[T(n,k,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 01 2022

%Y Cf. A001263 (m=0), A155467 (m=1), A155491 (m=3), this sequence (m=4).

%Y Cf. A142459.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Jan 23 2009

%E Edited by _G. C. Greubel_, Apr 01 2022