|
%I #23 Sep 08 2022 08:45:22
%S 1,3,1,9,5,1,27,19,7,1,81,65,33,9,1,243,211,131,51,11,1,729,665,473,
%T 233,73,13,1,2187,2059,1611,939,379,99,15,1,6561,6305,5281,3489,1697,
%U 577,129,17,1,19683,19171,16867,12259,6883,2851,835,163,19,1,59049,58025
%N Triangle read by rows: T(n,k) = Sum_{j=0..n} binomial(n, k+j)*2^(n-k-j).
%C T(n, 0) = A000244(n), T(n, 1) = A001047(n), T(n, 2) = A066810(n).
%C Column 0 is the row sums of A038207 starting at column 0, column 1 is the row sums of A038207 starting at column 1 etc. etc. Helpful suggestions related to Riordan arrays given by Paul Barry.
%C Riordan array ( 1/(1 - 3*x), x/(1 - 2*x) ). Matrix inverse is a signed version of A209149. - _Peter Bala_, Jul 17 2013
%C T(n,k) is the number of strings of length n over an alphabet of 3 letters that contain a given string of length k as a subsequence. - _Robert Israel_, Jan 14 2020
%H G. C. Greubel, <a href="/A112626/b112626.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, k) = Sum_{j=0..n} binomial(n, k+j)*2^(n-k-j).
%F O.g.f. (by columns): x^k /((1-3*x)*(1-2*x)^k). - Frank Ruskey and class
%F T(n,k) = Sum_{j=k..n} binomial(n,j)*2^(n-j). - _Ross La Haye_, May 02 2006
%F Binomial transform (by columns) of A055248.
%e Triangle begins as:
%e 1;
%e 3, 1;
%e 9, 5, 1;
%e 27, 19, 7, 1;
%e 81, 65, 33, 9, 1;
%e 243, 211, 131, 51, 11, 1;
%e 729, 665, 473, 233, 73, 13, 1...
%p seq(seq( add(binomial(n,j)*2^(n-j), j=k..n), k=0..n), n=0..10); # _G. C. Greubel_, Nov 18 2019
%t Flatten[Table[Sum[Binomial[n, k+m]*2^(n-k-m), {m, 0, n}], {n, 0, 10}, {k, 0, n}]]
%o (PARI) T(n,k) = sum(j=k,n, binomial(n,j)*2^(n-j)); \\ _G. C. Greubel_, Nov 18 2019
%o (Magma) [&+[Binomial(n,j)*2^(n-j): j in [k..n]]: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 18 2019
%o (Sage) [[sum(binomial(n,j)*2^(n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 18 2019
%o (GAP) Flat(List([0..10], n-> List([0..n], k-> Sum([k..n], j-> Binomial(n, j)*2^(n-j)) ))); # _G. C. Greubel_, Nov 18 2019
%Y Row sums = n*3^(n-1) + 3^n = A006234(n+3) (Frank Ruskey and class).
%Y Cf. A000244, A001047, A038207, A066810.
%Y Cf. A209149 (unsigned matrix inverse).
%K nonn,tabl
%O 0,2
%A _Ross La Haye_, Dec 26 2005
%E More terms from _Ross La Haye_, Dec 31 2006
|
