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A124644
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Triangle read by rows. T(n, k) = binomial(n, k) * CatalanNumber(n - k).
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8
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1, 1, 1, 2, 2, 1, 5, 6, 3, 1, 14, 20, 12, 4, 1, 42, 70, 50, 20, 5, 1, 132, 252, 210, 100, 30, 6, 1, 429, 924, 882, 490, 175, 42, 7, 1, 1430, 3432, 3696, 2352, 980, 280, 56, 8, 1, 4862, 12870, 15444, 11088, 5292, 1764, 420, 72, 9, 1, 16796, 48620, 64350, 51480, 27720
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OFFSET
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0,4
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COMMENTS
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Exponential Riordan array [exp(2x)*(Bessel_I(0,2x)-Bessel_I(1,2x)),x]. - Paul Barry, Mar 03 2011
O.g.f: G(x,t) = C[Pinv(x,t)] = {1 - sqrt[1 - 4 *x /(1-x*t)]}/2 where C(x) = [1 - sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108 with inverse Cinv(x) = x*(1-x), and Pinv(x,t)= -P(-x,t) = x/(1-t*x) with inverse P(x,t) = 1/(1+t*x). This puts this array in a family of arrays formed from the composition of C and P and their inverses. -G(-x,t) is the comp. inverse of the o.g.f. of A030528.
This is an Appell sequence with lowering operator d/dt p(n,t) = n*p(n-1,t) and (p(.,t)+a)^n = p(n,t+a). The e.g.f. has the form e^(x*t)/w(t) where 1/w(t) is the e.g.f. of the first column, which is the Catalan sequence A000108. (End)
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LINKS
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FORMULA
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T(n,k) = [x^(n-k)]F(-n,n-k+1;1;-1-x). - Paul Barry, Sep 05 2008
G.f.: 1/(1-xy-x/(1-x/(1-xy-x/(1-x/(1-xy-x/(1-x.... (continued fraction). - Paul Barry, Jan 06 2009
G.f.: 1/(1-x-xy-x^2/(1-2x-xy-x^2/(1-2x-xy-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
T(n,k) = Sum_{i = 0..n} C(n,i)*(-1)^(n-i)*Sum{j = 0..i} C(j,k)*C(i,j)*A000108(i-j). - Paul Barry, Aug 03 2009
Sum_{k = 0..n} T(n,k)*x^k = A126930(n), A005043(n), A000108(n), A007317(n+1), A064613(n), A104455(n) for x = -2, -1, 0, 1, 2, 3 respectively. T(n,k)= A007318(n,k)*A000108(n-k). - Philippe Deléham, Dec 12 2009
E.g.f.: exp(2*x + x*y)*(Bessel_I(0,2*x) - Bessel_I(1,2*x)). - Paul Barry, Mar 10 2010
O.g.f.: G(x,t) = C[P(x,t)] = [1 - sqrt(1-4*x / (1-t*x))] / 2 = Sum_{n >= 1} (C. + t)^(n-1) * x^n] = x + (1 + t) x^2 + (2 + 2t + t^2) x^3 + ... umbrally, where (C.)^n = C_n = (1,1,2,5,8,...) = A000108(x), C(x)= x*A000108(x)= G(x,0), and P(x,t) = x/(1 + t*x), a special linear fractional (Mobius) transformation. P(x,-t)= -P(-x,t) is the inverse of P(x,t).
Inverse o.g.f.: Ginv(x,t) = P[Cinv(x),-t] = x*(1-x) / [1 - t*x(1-x)] = -A030528(-x,t), where Cinv(x) = x*(1-x) is the inverse of C(x).
T(n, k) = (-1)^(n-k)*Catalan(n-k)*Pochhammer(-n,n-k)/(n-k)!. - Peter Luschny, Feb 05 2015
Recurrence: T(n, 0) = Catalan(n) = 1/(n+1)*binomial(2*n, n) and, for 1 <= k <= n, T(n, k) = (n/k) * T(n-1, k-1). - Peter Bala, Feb 04 2024
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EXAMPLE
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Triangle begins
1,
1, 1,
2, 2, 1,
5, 6, 3, 1,
14, 20, 12, 4, 1,
42, 70, 50, 20, 5, 1 (End)
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MAPLE
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m:=n->binomial(2*n, n)/(n+1): T:=proc(n, k) if k<=n then binomial(n, k)*m(n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od;
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MATHEMATICA
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Table[Binomial[n, #] Binomial[2 #, #]/(# + 1) &[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* or *)
Table[Abs[(-1)^k*CatalanNumber[#] Pochhammer[-n, #]/#!] &[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)
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PROG
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(Sage)
return (-1)^(n-k)*catalan_number(n-k)*rising_factorial(-n, n-k)/factorial(n-k)
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CROSSREFS
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KEYWORD
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AUTHOR
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Farkas Janos Smile (smile_farkasjanos(AT)yahoo.com.au), Dec 21 2006
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EXTENSIONS
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Name brought in line with the Maple program by Peter Luschny, Jun 21 2023
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STATUS
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approved
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