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A108073 Triangle in A071943 with rows reversed. 0
1, 1, 1, 3, 2, 1, 9, 7, 3, 1, 31, 24, 12, 4, 1, 113, 89, 46, 18, 5, 1, 431, 342, 183, 76, 25, 6, 1, 1697, 1355, 741, 323, 115, 33, 7, 1, 6847, 5492, 3054, 1376, 520, 164, 42, 8, 1, 28161, 22669, 12768, 5900, 2326, 786, 224, 52, 9, 1, 117631, 94962, 54033, 25464 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
A convolution triangle based on A052709 (with first term omitted). - Philippe Deléham, Sep 15 2005
LINKS
FORMULA
G.f.: (1-q)/(z(2 - t + 2z + tq)), where q = sqrt(1 - 4z - 4z^2). - Emeric Deutsch, Jun 06 2005
T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n; T(n, k) = Sum_{j>=0} T(n-1, k-1+j) + Sum_{j>=0} T(n-1, k+1+j). - Philippe Deléham, Sep 15 2005
T(n,k) = k*Sum_{i=1..(n-k)} C(i,n-k-i)*C(k+2*i-1,i)/(k+i), n > k, T(n,n)=1. - Vladimir Kruchinin, Apr 27 2015
EXAMPLE
1; 1,1; 3,2,1; 9,7,3,1; 31,24,12,4,1; ...
MAPLE
q:=sqrt(1-4*z-4*z^2): G:=(1-q)/z/(2-t+2*z+t*q): Gserz:=simplify(series(G, z=0, 14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gserz, z^n) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form # Emeric Deutsch, Jun 06 2005
MATHEMATICA
T[n_, n_] = 1; T[n_, k_] := (k+1)*Sum[Binomial[i, n-k-i] * Binomial[k+2*i, i] / (k+i+1), {i, 1, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Vladimir Kruchinin *)
PROG
(Maxima)
T(n, k):=if n=k then 1 else k*sum((binomial(i, n-k-i)*binomial(k+2*i-1, i))/(k+i), i, 1, n-k); /* Vladimir Kruchinin, Apr 27 2015 */
CROSSREFS
Row sums yield A071356. Column 0 yields A052709.
Sequence in context: A193918 A298804 A155788 * A057731 A126074 A108916
KEYWORD
nonn,tabl,easy
AUTHOR
N. J. A. Sloane, Jun 05 2005
EXTENSIONS
More terms from Emeric Deutsch, Jun 06 2005
STATUS
approved

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Last modified April 25 10:34 EDT 2024. Contains 371967 sequences. (Running on oeis4.)