|
%I #39 Dec 30 2023 23:50:28
%S 1,1,1,3,4,1,9,15,7,1,27,54,36,10,1,81,189,162,66,13,1,243,648,675,
%T 360,105,16,1,729,2187,2673,1755,675,153,19,1,2187,7290,10206,7938,
%U 3780,1134,210,22,1,6561,24057,37908,34020,19278,7182,1764,276,25,1
%N Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].
%C 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
%C Equals A080419 when first column is removed (here). - _Georg Fischer_, Jul 25 2023
%H G. C. Greubel, <a href="/A136158/b136158.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Sum_{k=0..n} T(n, k) = A081294(n).
%F 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.
%F T(n, k) = A038763(n,n-k). - _Philippe Deléham_, Dec 17 2007
%F 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
%F Sum_{k=0..n} T(n, k)*x^k = (1+x)*(3+x)^(n-1), n >= 1. - _Philippe Deléham_, Oct 30 2013
%F G.f.: (1-2*x)/(1-3*x-x*y). - _R. J. Mathar_, Aug 11 2015
%F From _G. C. Greubel_, Dec 22 2023: (Start)
%F T(n, 0) = A133494(n).
%F T(n, 1) = A006234(n+2).
%F T(n, 2) = A080420(n-2).
%F T(n, 3) = A080421(n-3).
%F T(n, 4) = A080422(n-4).
%F T(n, 5) = A080423(n-5).
%F T(n, n) = A000012(n).
%F T(n, n-1) = A016777(n-1).
%F T(n, n-2) = A062741(n-1).
%F Sum_{k=0..n} (-1)^k * T(n, k) = 0^n = A000007(n).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A003688(n).
%F Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A001519(n). (End)
%F From _G. C. Greubel_, Dec 27 2023: (Start)
%F T(n, k) = 3^(n-k-1)*(n+2*k)*binomial(n,k)/n, for n > 0, with T(0, 0) = 1.
%F T(n, k) = (-1)^k * A164948(n, k). (End)
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 3, 4, 1;
%e 9, 15, 7, 1;
%e 27, 54, 36, 10, 1;
%e 81, 189, 162, 66, 13, 1;
%e 243, 648, 675, 360, 105, 16, 1;
%e 729, 2187, 2673, 1755, 675, 153, 19, 1;
%e ...
%t A136158[n_,k_]:= If[n==0, 1, 3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
%t Table[A136158[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 22 2023; Dec 27 2023 *)
%o (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);
%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jul 25 2023
%o (Magma)
%o A136158:= func< n,k | n eq 0 select 1 else 3^(n-k-1)*(n+2*k)* Binomial(n, k)/n >;
%o [A136158(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 22 2023; Dec 27 2023
%o (SageMath)
%o def A136158(n,k): return 1 if (n==0) else 3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
%o flatten([[A136158(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Dec 22 2023; Dec 27 2023
%Y Cf. A000007, A000012, A001519, A003688, A006234, A016777, A062741.
%Y Cf. A080419, A080421, A080422, A080423, A081294, A133494, A136157.
%Y Absolute value of A164948.
%K nonn,tabl
%O 0,4
%A _Gary W. Adamson_, Dec 16 2007
%E More terms from _Philippe Deléham_, Dec 17 2007
|
