A116412 - OEIS

%I #37 Oct 29 2022 16:06:59

%S 1,3,1,6,6,1,12,21,9,1,24,60,45,12,1,48,156,171,78,15,1,96,384,558,

%T 372,120,18,1,192,912,1656,1473,690,171,21,1,384,2112,4608,5160,3225,

%U 1152,231,24,1,768,4800,12240,16584,13083,6219,1785,300,27,1,1536,10752

%N Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

%C Row sums are A003688. Diagonal sums are A116413. Product of A007318 and A116413 is A116414. Product of A007318 and A105475.

%C Subtriangle of triangle given by  (0, 3, -1, 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_, Jan 18 2012

%H Michael De Vlieger, <a href="/A116412/b116412.txt">Table of n, a(n) for n = 0..11475</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.

%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.

%F Number triangle T(n,k)=sum{j=0..n, C(k+1,j)*C(n-j,k)2^(n-k-j)}

%F From _Vladimir Kruchinin_, Mar 17 2011: (Start)

%F T((m+1)*n+r-1, m*n+r-1) * r/(m*n+r) = sum(k=1..n, k/n * T((m+1)*n-k-1, m*n-1) * T(r+k-1,r-1)), n>=m>1.

%F T(n-1,m-1) = m/n * sum(k=1..n-m+1, k*A003945(k-1)*T(n-k-1,m-2)), n>=m>1. (End)

%F G.f.: (1+x)/(1-(y+2)*x -y*x^2). - _Philippe Deléham_, Jan 18 2012

%F Sum_{k, 0<=k<=n} T(n,k)*x^k = A104537(n), A110523(n), (-2)^floor(n/2), A057079(n), A003945(n), A003688(n+1), A123347(n), A180035(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively. - _Philippe Deléham_, Jan 18 2012

%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(2,0) = T(2,1) = 6, T(2,2) = 1, T(n,k) = 0 if k>n or if k<0. - _Philippe Deléham_, Oct 31 2013

%e Triangle begins

%e 1,

%e 3, 1,

%e 6, 6, 1,

%e 12, 21, 9, 1,

%e 24, 60, 45, 12, 1,

%e 48, 156, 171, 78, 15, 1

%e Triangle T(n,k), 0<=k<=n, given by (0, 3, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins :

%e 1

%e 0, 1

%e 0, 3, 1

%e 0, 6, 6, 1

%e 0, 12, 21, 9, 1

%e 0, 24, 60, 45, 12, 1

%e 0, 48, 156, 171, 78, 15, 1

%e ... - _Philippe Deléham_, Jan 18 2012

%t With[{n = 10}, DeleteCases[#, 0] & /@ CoefficientList[Series[(1 + x)/(1 - (y + 2) x - y x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* _Michael De Vlieger_, Apr 25 2018 *)

%Y Cf. A003688, A003945.

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Feb 13 2006