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A071921 Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=1 by definition, read by antidiagonals.

%I A071921 #25 Dec 27 2023 18:05:24
%S A071921 1,1,0,1,1,0,1,2,1,0,1,3,4,1,0,1,4,9,7,1,0,1,5,16,22,11,1,0,1,6,25,50,
%T A071921 46,16,1,0,1,7,36,95,130,86,22,1,0,1,8,49,161,295,296,148,29,1,0,1,9,
%U A071921 64,252,581,791,610,239,37,1,0
%N A071921 Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=1 by definition, read by antidiagonals.
%C A071921 If one uses a definition of unimodality that involves universal quantifiers on the domain of a function then a(0,m)=1 a priori.
%H A071921 Alois P. Heinz, <a href="/A071921/b071921.txt">Antidiagonals n = 0..140, flattened</a>
%H A071921 Kenneth Edwards and Michael A. Allen, <a href="https://arxiv.org/abs/2009.04649">New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile</a>, arXiv:2009.04649 [math.CO], 2020.
%F A071921 a(n,m) = 1 if n=0, m>=0, a(n,m) = Sum_{k=0..m-1} C(2k+n-1,2k) otherwise.
%e A071921 Square array a(n,m) begins:
%e A071921   1, 1,  1,   1,    1,    1,     1,     1,      1, ...
%e A071921   0, 1,  2,   3,    4,    5,     6,     7,      8, ...
%e A071921   0, 1,  4,   9,   16,   25,    36,    49,     64, ...
%e A071921   0, 1,  7,  22,   50,   95,   161,   252,    372, ...
%e A071921   0, 1, 11,  46,  130,  295,   581,  1036,   1716, ...
%e A071921   0, 1, 16,  86,  296,  791,  1792,  3612,   6672, ...
%e A071921   0, 1, 22, 148,  610, 1897,  4900, 11088,  22716, ...
%e A071921   0, 1, 29, 239, 1163, 4166, 12174, 30738,  69498, ...
%e A071921   0, 1, 37, 367, 2083, 8518, 27966, 78354, 194634, ...
%p A071921 a:= (n, m)-> `if`(n=0, 1, add(binomial(n+2*j-1, 2*j), j=0..m-1)):
%p A071921 seq(seq(a(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Sep 22 2013
%t A071921 a[0, 0] = 1; a[n_, m_] := Sum[Binomial[2k+n-1, 2k], {k, 0, m-1}]; Table[a[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 11 2015 *)
%Y A071921 Cf. A071920, A225010.
%Y A071921 Main diagonal gives A088536 (for n>=1).
%K A071921 nonn,easy,tabl
%O A071921 0,8
%A A071921 Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002