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A098474
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A Catalan scaled binomial matrix.
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5
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1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 20, 14, 1, 5, 20, 50, 70, 42, 1, 6, 30, 100, 210, 252, 132, 1, 7, 42, 175, 490, 882, 924, 429, 1, 8, 56, 280, 980, 2352, 3696, 3432, 1430, 1, 9, 72, 420, 1764, 5292, 11088, 15444, 12870, 4862, 1, 10, 90, 600, 2940, 10584, 27720
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are A007317. Diagonal sums are A090344. Principal diagonal is A000108.
Comments from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 01 2005:
"Table U(n,k),k>=0, n>=0, read by antidiagonals, begins:
"row k = 0 : 1, 1, 2, 5, 14, 42, ... is A000108
"row k = 1 : 1, 2, 6, 20, 70, ... is A000984
"row k = 2 : 1, 3, 12, 50, 280, ... is A007854
"row k = 3 : 1, 4, 20, 104, 548, ... is A076035
"row k = 4 : 1, 5, 30, 185, 1150, ... is A076036
"...........................................
"G.f. for row k : 1/(1-(k+1)*x*C(x)) where C(x) is the g.f. = for Catalan numbers A000108.
"U(n,k) = sum_{j, 0<=j<=n} A106566(n,j)*(k+1)^j."
Mirror image of A124644 formatted as a triangular array. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 12 2009]
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FORMULA
| Number triangle T(n, k)=binomial(2k, k)binomial(n, k)/(k+1)
G.f.: 2/(1-x+(1-x-4*x*y)^(1/2)). E.g.f.: exp(x*(1+2*y))*(BesselI(0, 2*x*y)-BesselI(1, 2*x*y)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 11 2004
G.f.: 1/(1-x-xy/(1-xy/(1-x-xy/(1-xy/(1-x-xy/(1-xy/(1-x-xy/(1-xy/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 11 2009]
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A126930(n), A005043(n), A000108(n), A007317(n+1), A064613(n), A104455(n) for x = -2, -1, 0, 1, 2, 3 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 12 2009]
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EXAMPLE
| Rows begin {1}, {1,1}, {1,2,2}, {1,3,6,5}, {1,4,12,20,14},...
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CROSSREFS
| Cf. A098473.
Sequence in context: A082137 A091187 A065173 * A153199 A056860 A158825
Adjacent sequences: A098471 A098472 A098473 * A098475 A098476 A098477
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 09 2004
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