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 A060098 Triangle of partial sums of column sequences of triangle A060086, read by rows. 7
 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, 50, 71, 55, 26, 7, 1, 1, 20, 80, 140, 140, 86, 34, 8, 1, 1, 25, 120, 259, 316, 246, 126, 43, 9, 1, 1, 30, 175, 448, 660, 622, 399, 176, 53, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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). Row sums give A052534. Column sequences (without leading zeros) give A000012 (powers of 1), A002620(n+1), A002624, A060099-A060101 for m=0..5. The bisections of the column sequences give triangles A060102 and A060556. Riordan array (1/(1-x), x/((1-x)*(1-x^2))). - Paul Barry, Mar 28 2011 LINKS Vincenzo Librandi, Rows n = 0..100, flattened Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4. FORMULA 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)). 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 From Philippe Deléham, Apr 20 2023: (Start) T(n, k) = 0 if k < 0 or if k > n; T(n, k) = 1 if k = 0 or k = n; otherwise: T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k). T(n, k) = A188316(n, k) + A188316(n-1, k). (End) EXAMPLE p(3,x) = 1 + 4*x + 3*x^2 + x^3. Triangle begins: 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, 50, 71, 55, 26, 7, 1; ... MAPLE 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: seq(seq(A060098(n, k), k=0..n), n=0..12); # R. J. Mathar, Mar 29 2011 # Recurrence after Philippe Deléham: T := proc(n, k) option remember; if k < 0 or k > n then 0 elif k = 0 or n = k then 1 else T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k) fi end: for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, May 07 2023 MATHEMATICA 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 *) CROSSREFS Cf. A052534, A000012, A002620(n+1), A002624, A060099, A060100, A060101. Cf. A060102, A060556, A188316. Sequence in context: A026769 A257365 A230858 * A161492 A177976 A034781 Adjacent sequences: A060095 A060096 A060097 * A060099 A060100 A060101 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Apr 06 2001 STATUS approved

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Last modified May 20 23:58 EDT 2024. Contains 372720 sequences. (Running on oeis4.)