A306547 - OEIS

%I #15 Apr 19 2019 11:02:57

%S 1,1,-2,1,-11,4,1,-55,35,-8,1,-274,210,-91,16,1,-1368,986,-637,219,

%T -32,1,-6837,3180,-3473,1752,-507,64,1,-34181,-1431,-17951,10543,

%U -4563,1147,-128,1,-170900,-145310,-129950,48442,-30524,11470,-2555,256,1,-854494,-1726360,-1490890,-2314,-177832,84176,-28105,5627,-512

%N Triangle read by rows, defined by Riordan's general Eulerian recursion: T(n, k) = (k+3)*T(n-1, k) + (n-k-2) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).

%C Row sums are {1, -1, -6, -27, -138, -831, -5820, -46563, -419070, -4190703, ...}.

%C The Mathematica code for e(n,k,m) gives eleven sequences of which the first few are in the OEIS (see Crossrefs section).

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215.

%H G. C. Greubel, <a href="/A306547/b306547.txt">Rows n = 1..100 of triangle, flattened</a>

%F T(n, k) = (k+3)*T(n-1, k) + (n-k-2)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).

%F e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.

%e Triangle begins with:

%e 1.

%e 1,      -2.

%e 1,     -11,       4.

%e 1,     -55,      35,      -8.

%e 1,    -274,     210,     -91,    16.

%e 1,   -1368,     986,    -637,   219,    -32.

%e 1,   -6837,    3180,   -3473,  1752,   -507,    64.

%e 1,  -34181,   -1431,  -17951, 10543,  -4563,  1147,  -128.

%e 1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256.

%t e[n_, 0, m_]:= 1; (* Example for m=3 *)

%t e[n_, k_, m_]:= 0 /; k >= n;

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

%t Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]

%t T[n_, 1]:= 1; T[n_, n_]:= (-2)^(n-1); T[n_, k_]:= T[n, k] = (k+3)*T[n-1, k] + (n-k-2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]//Flatten

%o (PARI) {T(n, k) = if(k==1, 1, if(k==n, (-2)^(n-1), (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)))};

%o for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")))

%o (Sage)

%o def T(n, k):

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

%o     elif (k==n): return (-2)^(n-1)

%o     else: return (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)

%o [[T(n, k) for k in (1..n)] for n in (1..12)]

%Y Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3), this sequence.

%K sign,tabl

%O 1,3

%A _G. C. Greubel_, Feb 22 2019