This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A060098 #27 May 07 2023 18:27:41
%S A060098 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 A060098 50,71,55,26,7,1,1,20,80,140,140,86,34,8,1,1,25,120,259,316,246,126,
%U A060098 43,9,1,1,30,175,448,660,622,399,176,53,10,1
%N A060098 Triangle of partial sums of column sequences of triangle A060086, read by rows.
%C A060098 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 A060098 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 A060098 The bisections of the column sequences give triangles A060102 and A060556.
%C A060098 Riordan array (1/(1-x), x/((1-x)*(1-x^2))). - _Paul Barry_, Mar 28 2011
%H A060098 Vincenzo Librandi, <a href="/A060098/b060098.txt">Rows n = 0..100, flattened</a>
%H A060098 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 A060098 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 A060098 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 A060098 From _Philippe Deléham_, Apr 20 2023: (Start)
%F A060098 T(n, k) = 0 if k < 0 or if k > n; T(n, k) = 1 if k = 0 or k = n; otherwise:
%F A060098 T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k).
%F A060098 T(n, k) = A188316(n, k) + A188316(n-1, k). (End)
%e A060098 p(3,x) = 1 + 4*x + 3*x^2 + x^3.
%e A060098 Triangle begins:
%e A060098 1;
%e A060098 1, 1;
%e A060098 1, 2, 1;
%e A060098 1, 4, 3, 1;
%e A060098 1, 6, 8, 4, 1;
%e A060098 1, 9, 16, 13, 5, 1;
%e A060098 1, 12, 30, 32, 19, 6, 1;
%e A060098 1, 16, 50, 71, 55, 26, 7, 1;
%e A060098 ...
%p A060098 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 A060098 seq(seq(A060098(n,k), k=0..n), n=0..12); # _R. J. Mathar_, Mar 29 2011
%p A060098 # Recurrence after _Philippe Deléham_:
%p A060098 T := proc(n, k) option remember;
%p A060098 if k < 0 or k > n then 0 elif k = 0 or n = k then 1 else
%p A060098 T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k) fi end:
%p A060098 for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, May 07 2023
%t A060098 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 A060098 Cf. A052534, A000012, A002620(n+1), A002624, A060099, A060100, A060101.
%Y A060098 Cf. A060102, A060556, A188316.
%K A060098 nonn,easy,tabl
%O A060098 0,5
%A A060098 _Wolfdieter Lang_, Apr 06 2001