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A124644 Mirror image of A098474 formatted as a triangular array. 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Equal to A091867*A007318. - Philippe Deléham, Dec 12 2009

Exponential Riordan array [exp(2x)*(Bessel_I(0,2x)-Bessel_I(1,2x)),x]. - Paul Barry, Mar 03 2011

From Tom Copeland, Nov 04 2014: (Start)

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)

LINKS

Indranil Ghosh, Rows 0..125, flattened

P. Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6

FORMULA

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<=k<=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(2x+xy)*(Bessel_I(0,2x)-Bessel_I(1,2x)). - Paul Barry, Mar 10 2010

From Tom Copeland, Nov 08 2014: (Start)

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).

G(x,t) = x*A091867(x,t+1), and Ginv(x,t) = x*A104597(x,-(t+1)). (End)

T(n, k) = (-1)^(n-k)*Catalan(n-k)*Pochhammer(-n,n-k)/(n-k)!. - Peter Luschny, Feb 05 2015

EXAMPLE

From Paul Barry, Jan 28 2009: (Start)

Triangle begins

1,

1, 1,

2, 2, 1,

5, 6, 3, 1,

14, 20, 12, 4, 1,

42, 70, 50, 20, 5, 1 (End)

MAPLE

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;

MATHEMATICA

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 *)

PROG

(Sage)

def A124644(n, k):

    return (-1)^(n-k)*catalan_number(n-k)*rising_factorial(-n, n-k)/factorial(n-k)

for n in range(7): [A124644(n, k) for k in (0..n)] # Peter Luschny, Feb 05 2015

CROSSREFS

Cf. A098474, A000108, A091867, A030528, A104597.

Sequence in context: A019710 A118806 A171670 * A259691 A056857 A175579

Adjacent sequences:  A124641 A124642 A124643 * A124645 A124646 A124647

KEYWORD

nonn,tabl

AUTHOR

Farkas Janos Smile (smile_farkasjanos(AT)yahoo.com.au), Dec 21 2006

STATUS

approved

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Last modified February 25 14:46 EST 2018. Contains 299654 sequences. (Running on oeis4.)