cp's OEIS Frontend

A157013 Riordan's general Eulerian recursion: T(n, k) = (k+2)*T(n-1, k) + (n-k-1) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1).

%I A157013 #12 May 25 2023 03:57:32
%S A157013 1,1,-1,1,-4,1,1,-15,5,-1,1,-58,10,-6,1,1,-229,-66,-26,7,-1,1,-912,
%T A157013 -1017,-288,23,-8,1,1,-3643,-8733,-4779,-415,-41,9,-1,1,-14566,-61880,
%U A157013 -63606,-17242,-1158,40,-10,1,1,-58257,-396796,-691036,-375118,-60990,-1956,-60,11,-1
%N A157013 Riordan's general Eulerian recursion: T(n, k) = (k+2)*T(n-1, k) + (n-k-1) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1).
%C A157013 Row sums are {1, 0, -2, -10, -52, -314, -2200, -17602, -158420, -1584202, ...}.
%C A157013 This recursion set doesn't seem to produce the Eulerian 2nd A008517.
%C A157013 The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - _G. C. Greubel_, Feb 22 2019
%D A157013 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215
%H A157013 G. C. Greubel, <a href="/A157013/b157013.txt">Rows n = 1..100 of triangle, flattened</a>
%F A157013 e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.
%F A157013 T(n, k) = (k+2)*T(n-1, k) + (n-k-1)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1). - _G. C. Greubel_, Feb 22 2019
%e A157013 Triangle begins with:
%e A157013 1.
%e A157013 1,     -1.
%e A157013 1,     -4,       1.
%e A157013 1,    -15,       5,      -1.
%e A157013 1,    -58,      10,      -6,       1.
%e A157013 1,   -229,     -66,     -26,       7,     -1.
%e A157013 1,   -912,   -1017,    -288,      23,     -8,     1.
%e A157013 1,  -3643,   -8733,   -4779,    -415,    -41,     9,   -1.
%e A157013 1, -14566,  -61880,  -63606,  -17242,  -1158,    40,  -10,   1.
%e A157013 1, -58257, -396796, -691036, -375118, -60990, -1956,  -60,  11,  -1.
%t A157013 e[n_, 0, m_]:= 1;
%t A157013 e[n_, k_, m_]:= 0 /; k >= n;
%t A157013 e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
%t A157013 Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
%t A157013 T[n_,1]:=1; T[n_,n_]:=(-1)^(n-1); T[n_,k_]:= T[n,k] = (k+2)*T[n-1,k] + (n-k-1)*T[n-1,k-1]; Table[T[n,k], {n,1,10}, {k,1,n}]//Flatten (* _G. C. Greubel_, Feb 22 2019 *)
%o A157013 (PARI) {T(n, k) = if(k==1, 1, if(k==n, (-1)^(n-1), (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)))};
%o A157013 for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ _G. C. Greubel_, Feb 22 2019
%o A157013 (Sage)
%o A157013 def T(n, k):
%o A157013     if (k==1): return 1
%o A157013     elif (k==n): return (-1)^(n-1)
%o A157013     else: return (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)
%o A157013 [[T(n, k) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, Feb 22 2019
%Y A157013 Cf. A008517.
%Y A157013 Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), this sequence (m=3).
%K A157013 sign,tabl
%O A157013 1,5
%A A157013 _Roger L. Bagula_, Feb 21 2009