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A071920 Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=0 for all m>=0, read by antidiagonals. 14
0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 9, 7, 1, 0, 0, 5, 16, 22, 11, 1, 0, 0, 6, 25, 50, 46, 16, 1, 0, 0, 7, 36, 95, 130, 86, 22, 1, 0, 0, 8, 49, 161, 295, 296, 148, 29, 1, 0, 0, 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 existential quantifiers on the domain of a function then a(0,m)=0 a priori.

LINKS

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

FORMULA

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

EXAMPLE

0,    0,    0,    0,    0,    0,     0,     0,      0, ...

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

seq(seq(a(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Sep 21 2013

MATHEMATICA

a[n_, m_] := Sum[Binomial[n+2*k-1, 2*k], {k, 0, m-1}]; a[0, _] = 0; Table[a[n-m, m], {n, 0, 10}, {m, n, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, Feb 25 2015 *)

CROSSREFS

Cf. A071921, A225010.

Main diagonal is A088536.

Sequence in context: A099423 A221515 A221984 * A306548 A320531 A065719

Adjacent sequences:  A071917 A071918 A071919 * A071921 A071922 A071923

KEYWORD

nonn,easy,tabl

AUTHOR

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

STATUS

approved

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Last modified November 14 22:45 EST 2019. Contains 329135 sequences. (Running on oeis4.)