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Triangle of partial sums of column sequences of triangle A060086, read by rows.
7

%I #27 May 07 2023 18:27:41

%S 1,1,1,1,2,1,1,4,3,1,1,6,8,4,1,1,9,16,13,5,1,1,12,30,32,19,6,1,1,16,

%T 50,71,55,26,7,1,1,20,80,140,140,86,34,8,1,1,25,120,259,316,246,126,

%U 43,9,1,1,30,175,448,660,622,399,176,53,10,1

%N Triangle of partial sums of column sequences of triangle A060086, read by rows.

%C In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is (1/(1-x*z/((1-z^2)*(1-z))))/(1-z).

%C Row sums give A052534. Column sequences (without leading zeros) give A000012 (powers of 1), A002620(n+1), A002624, A060099-A060101 for m=0..5.

%C The bisections of the column sequences give triangles A060102 and A060556.

%C Riordan array (1/(1-x), x/((1-x)*(1-x^2))). - _Paul Barry_, Mar 28 2011

%H Vincenzo Librandi, <a href="/A060098/b060098.txt">Rows n = 0..100, flattened</a>

%H Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.

%F G.f. for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/(1-x) = x^m/((1+x)^m*(1-x)^(2*m+1)).

%F Number triangle T(n,k) = Sum_{i=0..floor(n/2)} C(n-2*i,n-2*i-k)*C(k+i-1,i). - _Paul Barry_, Mar 28 2011

%F From _Philippe Deléham_, Apr 20 2023: (Start)

%F T(n, k) = 0 if k < 0 or if k > n; T(n, k) = 1 if k = 0 or k = n; otherwise:

%F T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k).

%F T(n, k) = A188316(n, k) + A188316(n-1, k). (End)

%e p(3,x) = 1 + 4*x + 3*x^2 + x^3.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 4, 3, 1;

%e 1, 6, 8, 4, 1;

%e 1, 9, 16, 13, 5, 1;

%e 1, 12, 30, 32, 19, 6, 1;

%e 1, 16, 50, 71, 55, 26, 7, 1;

%e ...

%p A060098 := proc(n,k) add( binomial(n-2*i,n-2*i-k)*binomial(k+i-1,i), i=0..floor(n/2)) ; end proc:

%p seq(seq(A060098(n,k), k=0..n), n=0..12); # _R. J. Mathar_, Mar 29 2011

%p # Recurrence after _Philippe Deléham_:

%p T := proc(n, k) option remember;

%p if k < 0 or k > n then 0 elif k = 0 or n = k then 1 else

%p T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k) fi end:

%p for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, May 07 2023

%t t[n_, k_] := Sum[ Binomial[n-2*j, n-2*j-k]*Binomial[k+j-1, j], {j, 0, n/2}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 21 2013 *)

%Y Cf. A052534, A000012, A002620(n+1), A002624, A060099, A060100, A060101.

%Y Cf. A060102, A060556, A188316.

%K nonn,easy,tabl

%O 0,5

%A _Wolfdieter Lang_, Apr 06 2001