A136158 Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].

1, 1, 1, 3, 4, 1, 9, 15, 7, 1, 27, 54, 36, 10, 1, 81, 189, 162, 66, 13, 1, 243, 648, 675, 360, 105, 16, 1, 729, 2187, 2673, 1755, 675, 153, 19, 1, 2187, 7290, 10206, 7938, 3780, 1134, 210, 22, 1, 6561, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1

Offset: 0

Author

Gary W. Adamson, Dec 16 2007

Keywords

• nonn
• tabl

Comments

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 17 2007

Equals A080419 when first column is removed (here). - Georg Fischer, Jul 25 2023

Examples

			First few rows of the triangle:
    1;
    1,    1;
    3,    4,    1;
    9,   15,    7,    1;
   27,   54,   36,   10,   1;
   81,  189,  162,   66,  13,   1;
  243,  648,  675,  360, 105,  16,  1;
  729, 2187, 2673, 1755, 675, 153, 19, 1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000007, A000012, A001519, A003688, A006234, A016777, A062741.

Cf. A080419, A080421, A080422, A080423, A081294, A133494, A136157.

Absolute value of A164948.

Programs

Magma

A136158:= func< n,k | n eq 0 select 1 else 3^(n-k-1)*(n+2*k)* Binomial(n, k)/n >;
[A136158(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 22 2023; Dec 27 2023

Mathematica

A136158[n_,k_]:= If[n==0, 1, 3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
Table[A136158[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2023; Dec 27 2023 *)

PARI

T(n,k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); 3*T(n-1,k) + T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 25 2023

SageMath

def A136158(n,k): return 1 if (n==0) else 3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
flatten([[A136158(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 22 2023; Dec 27 2023

Formula

Sum_{k=0..n} T(n, k) = A081294(n).

Given A136157 = M, an infinite lower triangular bidiagonal matrix with (3, 3, 3, ...) in the main diagonal, (1, 1, 1, ...) in the subdiagonal and the rest zeros; rows of A136157 are generated from M^n * [1, 1, 0, 0, 0, ...], given a(0) = 1.

T(n, k) = A038763(n,n-k). - Philippe Deléham, Dec 17 2007

T(n, k) = 3*T(n-1, k) + T(n-1, k-1) for n > 1, T(0,0) = T(1,1) = T(1,0) = 1. - Philippe Deléham, Oct 30 2013

Sum_{k=0..n} T(n, k)*x^k = (1+x)*(3+x)^(n-1), n >= 1. - Philippe Deléham, Oct 30 2013

G.f.: (1-2*x)/(1-3*x-x*y). - R. J. Mathar, Aug 11 2015

From G. C. Greubel, Dec 22 2023: (Start)

T(n, 0) = A133494(n).

T(n, 1) = A006234(n+2).

T(n, 2) = A080420(n-2).

T(n, 3) = A080421(n-3).

T(n, 4) = A080422(n-4).

T(n, 5) = A080423(n-5).

T(n, n) = A000012(n).

T(n, n-1) = A016777(n-1).

T(n, n-2) = A062741(n-1).

Sum_{k=0..n} (-1)^k * T(n, k) = 0^n = A000007(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A003688(n).

Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A001519(n). (End)

From G. C. Greubel, Dec 27 2023: (Start)

T(n, k) = 3^(n-k-1)*(n+2*k)*binomial(n,k)/n, for n > 0, with T(0, 0) = 1.

T(n, k) = (-1)^k * A164948(n, k). (End)

Extensions

More terms from Philippe Deléham, Dec 17 2007