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A099020 Euler-Seidel matrix T(k,n) with start sequence A001147, read by antidiagonals.

%I A099020 #15 Jul 06 2023 07:04:41
%S A099020 1,1,0,2,1,1,4,2,1,0,10,6,4,3,3,26,16,10,6,3,0,76,50,34,24,18,15,15,
%T A099020 232,156,106,72,48,30,15,0,764,532,376,270,198,150,120,105,105,2620,
%U A099020 1856,1324,948,678,480,330,210,105,0,9496,6876,5020,3696,2748,2070,1590,1260,1050,945,945
%N A099020 Euler-Seidel matrix T(k,n) with start sequence A001147, read by antidiagonals.
%C A099020 In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences, with the first column the inverse binomial transform of the start sequence.
%H A099020 Alois P. Heinz, <a href="/A099020/b099020.txt">Rows n = 0..140, flattened</a>
%H A099020 D. Dumont, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05dumont.html">Matrices d'Euler-Seidel</a>, Sem. Loth. Comb. B05c (1981) 59-78.
%F A099020 Recurrence: T(0, 2n) = (2n-1)!!, T(0, 2n+1) = 0, T(k, n) = T(k-1, n) + T(k-1, n+1).
%e A099020 1,   0,  1,  0,   3,   0,   15, ...
%e A099020 1,   1,  1,  3,   3,  15,   15, ...
%e A099020 2,   2,  4,  6,  18,  30,  120, ...
%e A099020 4,   6, 10, 24,  48, 150,  330, ...
%e A099020 10, 16, 34, 72, 198, 480, 1590, ...
%p A099020 T:= proc(k, n) option remember; `if`(k=0, `if`(irem(n, 2)=0,
%p A099020       doublefactorial(n-1), 0), T(k-1, n) +T(k-1, n+1))
%p A099020     end:
%p A099020 seq(seq(T(d-n, n), n=0..d), d=0..14);  # _Alois P. Heinz_, Oct 14 2012
%t A099020 t[0, n_?EvenQ] := (n-1)!!; t[0, n_?OddQ] := 0; t[k_, n_] := t[k, n] = t[k-1, n] + t[k-1, n+1]; Table[t[k-n, n], {k, 0, 10}, {n, 0, k}] // Flatten (* _Jean-François Alcover_, Dec 10 2012 *)
%Y A099020 First column is A000085, 2nd A013989, main diagonal is in A099021.
%K A099020 nonn,tabl
%O A099020 0,4
%A A099020 _Ralf Stephan_, Sep 23 2004