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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. 9
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, 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, 64, 252, 581, 791, 610, 239, 37, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

If one uses a definition of unimodality that involves universal quantifiers on the domain of a function then a(0,m)=1 a priori.

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

FORMULA

a(n,m) = 1 if n=0, m>=0, a(n,m) = Sum_{k=0..m-1} C(2k+n-1,2k) otherwise.

EXAMPLE

Square array a(n,m) begins:

1, 1,  1,   1,    1,    1,     1,     1,      1, ...

0, 1,  2,   3,    4,    5,     6,     7,      8, ...

0, 1,  4,   9,   16,   25,    36,    49,     64, ...

0, 1,  7,  22,   50,   95,   161,   252,    372, ...

0, 1, 11,  46,  130,  295,   581,  1036,   1716, ...

0, 1, 16,  86,  296,  791,  1792,  3612,   6672, ...

0, 1, 22, 148,  610, 1897,  4900, 11088,  22716, ...

0, 1, 29, 239, 1163, 4166, 12174, 30738,  69498, ...

0, 1, 37, 367, 2083, 8518, 27966, 78354, 194634, ...

MAPLE

a:= (n, m)-> `if`(n=0, 1, add(binomial(n+2*j-1, 2*j), j=0..m-1)):

seq(seq(a(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Sep 22 2013

MATHEMATICA

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-Fran├žois Alcover, Nov 11 2015 *)

CROSSREFS

Cf. A071920, A225010.

Sequence in context: A118340 A213276 A210391 * A003992 A246118 A171882

Adjacent sequences:  A071918 A071919 A071920 * A071922 A071923 A071924

KEYWORD

nonn,easy,tabl

AUTHOR

Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002

STATUS

approved

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Last modified December 10 09:49 EST 2018. Contains 318047 sequences. (Running on oeis4.)