A073370 - OEIS

%I #54 Oct 07 2022 07:33:07

%S 1,1,1,3,2,1,5,7,3,1,11,16,12,4,1,21,41,34,18,5,1,43,94,99,60,25,6,1,

%T 85,219,261,195,95,33,7,1,171,492,678,576,340,140,42,8,1,341,1101,

%U 1692,1644,1106,546,196,52,9,1

%N Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.

%C The g.f. for the row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(1-(1+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.

%C Riordan array (1/(1-x-2*x^2), x/(1-x-2*x^2)). - _Paul Barry_, Mar 15 2005

%C Subtriangle (obtained by dropping the first column) of the triangle given by (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 19 2013

%C The number of ternary words of length n having k letters equal 2 and 0,1 avoid runs of odd lengths. - _Milan Janjic_, Jan 14 2017

%H G. C. Greubel, <a href="/A073370/b073370.txt">Rows n = 0..50 of the triangle, flattened</a>

%H W. Lang, <a href="/A073370/a073370.txt">First 10 rows</a>.

%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.

%F T(n, m) = Sum_{k=0..floor((n-m)/2)} binomial(n-k, m)*binomial(n-m-k, k)*2^k, if n > m, else 0.

%F Sum_{k=0..n} T(n, k) = A002605(n+1).

%F T(n, m) = (1*(n-m+1)*T(n, m-1) + 2*2*(n+m)*T(n-1, m-1))/((1^2 + 4*2)*m), n >= m >= 1, T(n, 0) = A001045(n+1), n >= 0, else 0.

%F T(n, m) = (p(m-1, n-m)*1*(n-m+1)*T(n-m+1) + q(m-1, n-m)*2*(n-m+2)*T(n-m))/(m!*9^m), n >= m >= 1, with T(n) = T(n, m=0) = A001045(n+1), else 0; p(k, n) = Sum_{j=0..k} (A(k, j)*n^(k-j) and q(k, n) = Sum_{j=0..k} B(k, j)*n^(k-j), with the number triangles A(k, m) = A073399(k, m) and B(k, m) = A073400(k, m).

%F G.f.: 1/(1-(1+2*x)*x)^(m+1) = 1/((1+x)*(1-2*x))^(m+1), m >= 0, for column m (without leading zeros).

%F T(n, 0) = A001045(n), T(1, 1) = 1, T(n, k) = 0 if k>n, T(n, k) = T(n-1, k-1) + 2*T(n-2, k) + T(n-1, k) otherwise. - _Paul Barry_, Mar 15 2005

%F G.f.: (1+x)*(1-2*x)/(1-x-2*x^2-x*y) for the triangle including the 1, 0, 0, 0, 0, ... column. - _R. J. Mathar_, Aug 11 2015

%F From _Peter Bala_, Oct 07 2019: (Start)

%F Recurrence for row polynomials: R(n,x) = (1 + x)*R(n-1,x) + 2*R(n-2,x) with R(0,x) = 1 and R(1,x) = 1 + x.

%F The row reverse polynomial x^n*R(n,1/x) is equal to the numerator polynomial of the finite continued fraction 1 + x/(1 - 2*x/(1 + ... + x/(1 - 2*x/(1)))) (with 2*n partial numerators). Cf. A110441. (End)

%F From _G. C. Greubel_, Oct 01 2022: (Start)

%F T(n, k) = binomial(n,k)*Sum_{j=0..floor((n-k)/2)} 2^j*binomial(2*j, j)*binomial(n-k, 2*j)/binomial(n, j).

%F T(n, k) = binomial(n, k)*Hypergeometric2F1([(k-n)/2, (k-n+1)/2], [-2*n], -8).

%F Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A006130(n).

%F Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000045(n+1). (End)

%e Triangle begins as:

%e     1;

%e     1,   1;

%e     3,   2,   1;

%e     5,   7,   3,   1;

%e    11,  16,  12,   4,   1;

%e    21,  41,  34,  18,   5,   1;

%e    43,  94,  99,  60,  25,   6,   1;

%e    85, 219, 261, 195,  95,  33,   7,   1;

%e   171, 492, 678, 576, 340, 140,  42,   8,   1;

%e The triangle (0, 1, 2, -2, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:

%e   1;

%e   0,  1;

%e   0,  1,  1;

%e   0,  3,  2,  1;

%e   0,  5,  7,  3,  1;

%e   0, 11, 16, 12,  4,  1;

%e   0, 21, 41, 34, 18,  5,  1; - _Philippe Deléham_, Feb 19 2013

%p # Uses function PMatrix from A357368. Adds a row above and a column to the left.

%p PMatrix(10, n -> (2^n - (-1)^n) / 3); # _Peter Luschny_, Oct 07 2022

%t T[n_, k_]:= T[n, k]= Sum[Binomial[n-j,k]*Binomial[n-k-j,j]*2^j, {j,0,Floor[(n- k)/2]}];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 01 2022 *)

%o (Magma)

%o A073370:= func< n,k | (&+[Binomial(n-j,k)*Binomial(n-k-j,j)*2^j: j in [0..Floor((n-k)/2)]]) >;

%o [A073370(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 01 2022

%o (SageMath)

%o def A073370(n,k): return binomial(n,k)*sum( 2^j * binomial(2*j,j) * binomial(n-k,2*j)/binomial(n,j) for j in range(1+(n-k)//2))

%o flatten([[A073370(n,k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Oct 01 2022

%Y Columns: A001045 (k=0), A073371 (k=1), A073372 (k=2), A073373 (k=3), A073374 (k=4), A073375 (k=5), A073376 (k=6), A073377 (k=7), A073378 (k=8), A073379 (k=9).

%Y Cf. A002605 (row sums), A006130 (diagonal sums), A073399, A073400.

%Y Cf. A000045, A077957.

%K nonn,easy,tabl

%O 0,4

%A _Wolfdieter Lang_, Aug 02 2002