A226031 -id:A226031 - cp's OEIS frontend

A227402 Number of unimodal functions f:[n]->[n^2].

1, 1, 16, 525, 24616, 1505205, 113772114, 10253539205, 1073769343504, 128165285630637, 17177527372642000, 2554518029816653175, 417444979902876203656, 74358489250362053975095, 14340040595865309129453250, 2976703788777987140216622005

Offset: 0

Alois P. Heinz, Sep 20 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..110

Main diagonal of A226031.

Cf. A071920, A227406.

a:= n-> `if`(n=0, 1, add(binomial(n+2*j-1, 2*j), j=0..n^2-1)):
seq(a(n), n=0..20);

Flatten[{1,Table[Sum[Binomial[n+2*j-1, 2*j], {j,0,n^2-1}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 29 2014 *)

a(n) = Sum_{j=0..n^2-1} C(n+2*j-1,2*j), a(0) = 1.
a(n) = A071921(n,n^2).
a(n) ~ 2^(n-3/2) * n^(n-1/2) * exp(n+1/4) / sqrt(Pi). - Vaclav Kotesovec, Aug 29 2014

A226012 Number of unimodal functions f:[n]->[2n].

Original entry on oeis.org

1, 2, 16, 161, 1716, 18832, 210574, 2385644, 27290916, 314537894, 3646709616, 42483615330, 496908084660, 5831654186256, 68636514069496, 809835178438996, 9575879777488676, 113445872396014898, 1346272950075766624, 16000494256911975827, 190424554847852203816

Offset: 0

Alois P. Heinz, May 22 2013

nonn

			a(2) = 16: [1,1], [1,2], [1,3], [1,4], [2,1], [2,2], [2,3], [2,4], [3,1], [3,2], [3,3], [3,4], [4,1], [4,2], [4,3], [4,4].

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=2 of A226031.

a:= proc(n) option remember; `if`(n<3, 2^(n^2),
      ((2166498*n^7 -16827434*n^6 +54145990*n^5 -93141070*n^4
       +92008232*n^3 -51863736*n^2 +15330240*n -1814400) *a(n-1)
      -5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)
        *(333*n^3-595*n^2+338*n-60) *a(n-2)) / (16*(4*n-3)*
         (2*n-1)*(4*n-5)*(333*n^3-1594*n^2+2527*n-1326)*n))
    end:
seq(a(n), n=0..30);

A[n_, k_] := If[n==0, 1, Sum[Binomial[n + 2j - 1, 2j], {j, 0, k n - 1}]];
a[n_] := A[n, 2];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz in A226031 *)

a(n) ~ 5^(5*n-1/2) / (9*2^(8*n-5/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jul 16 2014

cp's OEIS Frontend

A227402 Number of unimodal functions f:[n]->[n^2].

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