A377443 Triangular array T(n,k) read by rows, satisfies A377441(n, k+2) = Sum_{m=0..k} T(k, m)*n^m.

2, 5, 1, 14, 6, 1, 42, 27, 8, 1, 132, 111, 45, 10, 1, 429, 441, 222, 67, 12, 1, 1430, 1728, 1029, 382, 93, 14, 1, 4862, 6733, 4608, 2005, 599, 123, 16, 1, 16796, 26181, 20199, 10018, 3495, 881, 157, 18, 1, 58786, 101763, 87270, 48445, 19188, 5641, 1236, 195, 20, 1

Offset

0,1

Links

Table of n, a(n) for n=0..54.

Formula

G.f.: (-(y*x^3-(y+1)*x^2+2*x+1) + sqrt((y*x^3-(y+1)*x^2+x)^2 - 4*(x^3-x^2)*((y+1)*x^2-x)))/(2*(x^3-x^2))/x^2.

T(n, 0) = A000108(n+2).

T(n, 1) = A371965(n+2).

T(n, 2) G.f.: x^2*1/( (x - 1)^2*(1 - 4*x)^(3/2) ).

T(n, 3) G.f.: x^3*(3*x - 1)/( (x - 1)^3*(1 - 4*x)^(5/2) ).

T(n, 4) G.f.: x^4*(x^3 + (3*x - 1)^2)/( (x - 1)^4*(1 - 4*x)^(7/2) ).

T(n, 5) G.f.: x^5*(3*x^3*(3*y - 1) + (3*x - 1)^3)/( (x - 1)^5*(1 - 4*x)^(9/2) ).

T(n, 6) G.f.: x^6*(2*x^6 + 6*x^3*(3*x - 1)^2 + (3*x - 1)^4)/( (x - 1)^6*(1 - 4*x)^(11/2) ) ).

T(n, 7) G.f.: x^7*(10*x^6*(3*x - 1) + 10*x^3*(3*x - 1)^3 + (3*x - 1)^5)/( (x - 1)^7*(1 - 4*x)^(13/2) ).

0 = Sum_{n=0..k} T(n+k, n)*(-1)^n*binomial(k, n).

The diagonal k terms below main diagonal has G.f.: 1 + Sum_{m=1..k+1} A175136(k+2, k-m+2)*(1 - x)^k.

T(n+k, n) = Sum_{m=1..k+1} A175136(k+2, k-m+2)*binomial(m+n-1, m-1), for k > 0.

Example

Triangle T(n, k) starts:
[0]     2
[1]     5,      1
[2]    14,      6,     1
[3]    42,     27,     8,     1
[4]   132,    111,    45,    10,     1
[5]   429,    441,   222,    67,    12,    1
[6]  1430,   1728,  1029,   382,    93,   14,    1
[7]  4862,   6733,  4608,  2005,   599,  123,   16,   1
[8] 16796,  26181, 20199, 10018,  3495,  881,  157,  18,  1
[9] 58786, 101763, 87270, 48445, 19188, 5641, 1236, 195, 20, 1

Prog

(PARI)
A377441(n, max_k) = Vec(-2*((n+1)*x-1)/((x-1)*(n*x-1)+((n*x^2-(n+1)*x+1)^2-4*x*(x-1)*((n+1)*x-1)+O(x^max_k))^(1/2)))
T(n, k) = Vec(A377441(y, n+5)[n+3])[n-k+1]
(PARI)
A091894(n, k) = 2^(n-2*k-1)*binomial(n-1, 2*k)*(binomial(2*k, k)/(k + 1))
A175136(n, k) = sum(m=0, (n - k)/2, A091894(n-k, m)*binomial(n-m-1, n-k))
T(n, k) = sum(m=1, n+1-k, A175136(n+2-k, n-m+2-k)*binomial(m+k-1, m-1))+(k==0)

Crossrefs

Cf. A377441, A377442.

Cf. A254316 (row sums).

Cf. A000108, A091894, A175136, A371965.

Sequence in context: A118964 A263771 A073187 * A138159 A118919 A319120

Adjacent sequences: A377440 A377441 A377442 * A377444 A377445 A377446

Keyword

nonn,tabl,new

Author

Thomas Scheuerle, Nov 04 2024

Status

approved