A133656 -id:A133656 - cp's OEIS frontend

A201080 Irregular triangle read by rows: number of shifted Schroeder paths of length n and area k.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 4, 3, 3, 3, 1, 1, 1, 1, 3, 3, 5, 6, 8, 9, 11, 12, 11, 10, 7, 6, 4, 1, 1, 1, 1, 3, 3, 6, 6, 9, 12, 16, 18, 22, 27, 29, 33, 38, 40, 39, 39, 34, 28, 21, 14, 10, 5, 1, 1, 1, 1, 3, 3, 6, 6, 10, 13, 18, 22, 28, 35, 41, 50, 61

Offset: 0

N. J. A. Sloane, Nov 26 2011

			Triangle begins
1
1 1
1 1 1 2 1
1 1 1 3 3 4 3 3 3 1
1 1 1 3 3 5 6 8 9 11 12 11 10 7 6 4 1
...

Alois P. Heinz, Rows n = 0..40, flattened
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953. See Example 3.

Row sums give A133656.

b:= proc(x, y) option remember; expand(`if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y)*z^(2*y)+b(x, y-1)+`if`(y>0, add(
       b(x-(2*j-1), y-1)*z^((2*y-1)*(2*j-1)), j=1..1+(x-y)/2), 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Feb 02 2018

b[x_, y_] := b[x, y] = Expand[If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y]*z^(2*y) + b[x, y - 1] + If[y > 0, Sum[b[x - (2*j - 1), y - 1]*z^((2*y - 1)*(2*j - 1)), {j, 1, 1 + (x - y)/2}], 0]]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, n]];
Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

More term from Alois P. Heinz, Feb 02 2018

A348474 Number of compositions of n into exactly 2n nonnegative parts such that each positive i-th part is odd if i is odd.

Original entry on oeis.org

1, 2, 8, 41, 220, 1212, 6803, 38691, 222196, 1285610, 7482718, 43762754, 256972507, 1514020484, 8945944435, 52990732161, 314568593860, 1870939233546, 11146516959176, 66508200091575, 397375460647690, 2377167144881136, 14236462650026064, 85346464443885086

Offset: 0

Alois P. Heinz, Oct 19 2021

nonn

			a(2) = 8: [0,0,0,2], [0,0,1,1], [0,1,0,1], [0,1,1,0], [0,2,0,0], [1,0,0,1], [1,0,1,0], [1,1,0,0].

Alois P. Heinz, Table of n, a(n) for n = 0..1272

Cf. A133656, A348410, A348476, A348478.

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(b(n-j, t-1)*iquo(j+3, 2), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n],
     Sum[b[n - j, t - 1]*Quotient[j + 3, 2], {j, 0, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 07 2022, after Alois P. Heinz *)

a(n) ~ c * d^n / sqrt(Pi*n), where d = 6.12846447590595003785095345916525... is the real root of the equation 32*d^4 - 195*d^3 + 12*d^2 - 112*d - 20 = 0 and c = 0.5667463795063214394117147185755881... is positive root of the equation 182464*c^8 - 45616*c^6 - 2108*c^4 - 601*c^2 - 20 = 0. - Vaclav Kotesovec, Nov 01 2021
From Peter Bala, Feb 22 2022: (Start)
Conjecture: a(n) = [x^n] ( (1 + x - x^2)/((1 + x)*(1 - x)^2) )^n.
If true, then the following hold:
a(n) = Sum_{i = 0..n} Sum_{j = 0..n} binomial(n,i+2*j)*binomial(2*i+2*j-1, i)*binomial(n+j-1,j).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 6*x^2 + 23*x^3 + 99*x^4 + ... is the g.f. of A133656.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)

cp's OEIS Frontend

A201080 Irregular triangle read by rows: number of shifted Schroeder paths of length n and area k.

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