cp's OEIS Frontend

A217536 Square array read by antidiagonals, where the top row is the nonnegative integers and the other numbers are the sum of the neighbors in the preceding row.

%I A217536 #52 Sep 20 2024 06:14:35
%S A217536 0,1,1,2,3,4,3,6,10,14,4,9,18,32,46,5,12,27,55,101,147,6,15,36,81,168,
%T A217536 315,462,7,18,45,108,244,513,975,1437,8,21,54,135,324,736,1564,3001,
%U A217536 4438,9,24,63,162,405,973,2222,4761,9199,13637,10,27,72,189,486,1215,2924,6710,14472,28109,41746
%N A217536 Square array read by antidiagonals, where the top row is the nonnegative integers and the other numbers are the sum of the neighbors in the preceding row.
%C A217536 Each number in the top row of the array is determined by the pre-defined sequence (in this case, the nonnegative integers). Each number in lower rows is the sum of the numbers vertically or diagonally above it (so, the number at the left end of each row is the sum of two numbers, and all other numbers the sum of three).
%C A217536 Replacing the top row with A000012 (the all 1's sequence) and constructing the rest of the array the same way produces A062105. Similarly, replacing the top row with A000007 (a(n) = 0^n) produces A020474. - _WG Zeist_, Aug 24 2024
%C A217536 For any array constructed with this method, regardless of the sequence chosen for the top row, the sequence in the first column of the array can be computed from the sequence in the top row as follows: let a(0), a(1), a(2), ... be the terms in the top row, and b(0), b(1), b(2), ... the terms in the first column. Then b(n) = Sum_{k=0..n} A064189(n,k) * a(k). The inverse operation, to compute the top row from the first column, is given by a(n) = Sum_{k=0..n} A104562(n,k) * b(k). - _WG Zeist_, Aug 26 2024
%H A217536 Alois P. Heinz, <a href="/A217536/b217536.txt">Antidiagonals n = 0..150, flattened</a>
%F A217536 T(m+1,n) = sum(T(m,k), |k-n| <= 1) (and T(0,n)=n), m, n >= 0. - _M. F. Hasler_, Oct 09 2012
%e A217536 The array starts:
%e A217536   0  1  2  3
%e A217536   1  3  6  9
%e A217536   4  10 18 27
%e A217536   14 32 55 81
%p A217536 A:= proc(n, k) option remember; `if`(k<0, 0,
%p A217536      `if`(n=0, k, add(A(n-1, k+i), i=-1..1)))
%p A217536     end:
%p A217536 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Aug 24 2024
%Y A217536 Main diagonal gives A036290. First column gives A330796.
%Y A217536 Cf. A020474, A062105, A064189, A104562.
%K A217536 nonn,tabl
%O A217536 0,4
%A A217536 _WG Zeist_, Oct 06 2012
%E A217536 Offset 0 from _Alois P. Heinz_, Aug 24 2024