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A098474 Triangle read by rows, T(n,k) = C(n,k)*C(2*k,k)/(k+1), n>=0, 0<=k<=n. 10
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; text; internal format)
OFFSET

0,5

COMMENTS

A Catalan scaled binomial matrix.

Row sums are A007317.

Diagonal sums are A090344.

Principal diagonal is A000108.

From Philippe Deléham, Sep 01 2005: (Start)

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. (End)

LINKS

Indranil Ghosh, Rows 0..125, flattened

FORMULA

G.f.: 2/(1-x+(1-x-4*x*y)^(1/2)). - Vladeta Jovovic, Sep 11 2004

E.g.f.: exp(x*(1+2*y))*(BesselI(0, 2*x*y)-BesselI(1, 2*x*y)). - Vladeta Jovovic, 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). - Paul Barry, 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. - Philippe Deléham, Dec 12 2009

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

O.g.f.: [1 - sqrt(1-4tx/(1-x))]/(2tx) = 1 + (1+t) x + (1+2t+2t^2) x^2 + (1+3t+6t^2+5t^3) x^3 + ... , generating the polynomials of this entry, reverse of A124644. See A011973 for a derivation and the inverse o.g.f., connected to the Fibonacci, Chebyshev, and Motzkin polynomials. See also A267633. - Tom Copeland, Jan 25 2016

From Peter Bala, Jun 13 2016: (Start)

The o.g.f. F(x,t) = ( 1 - sqrt(1 - 4*t*x/(1 - x)) )/(2*t*x) satisfies the partial differential equation d/dx(x*(1 - x)*F) - x*t*(1 + 4*t)*dF/dt - 2*x*t*F = 1. This gives a recurrence for the row polynomials: (n + 2)*R(n+1,t) = t*(1 + 4*t)*R'(n,t) + (2*t + n + 2)*R(n,t), where the prime ' indicates differentiation with respect to t.

Equivalently, setting Q(n,t) = t^(n+2)*R(n,-t)/(1 - 4*t)^(n + 3/2) we have t^2*d/dt(Q(n,t)) = (n + 2)*Q(n+1,t).

This leads to the following expansions:

Q(0,t) = 1/2*Sum_{k >= 1} k*binomial(2*k,k)*t^(k+1)

Q(1,t) = 1/2*Sum_{k >= 1} k*(k+1)/2!*binomial(2*k,k)*t^(k+2)

Q(2,t) = 1/2*Sum_{k >= 1} k*(k+1)*(k+2)/3!*binomial(2*k,k) *t^(k+3) and so on. (End)

EXAMPLE

Rows begin:

[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]

...

Row 3: t*(1 - 3*t + 6*t^2 - 5*t^3)/(1 - 4*t)^(9/2) = 1/2*Sum_{k >= 1} k*(k+1)*(k+2)*(k+3)/4!*binomial(2*k,k)*t^k. - Peter Bala, Jun 13 2016

MATHEMATICA

Table[Binomial[n, k] Binomial[2 k, k]/(k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* or *)

Table[(-1)^k*CatalanNumber[k] Pochhammer[-n, k]/k!, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

PROG

(Sage)

def A098474(n, k):

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

for n in range(7): [A098474(n, k) for k in (0..n)] # Peter Luschny, Feb 05 2015

CROSSREFS

Mirror image of A124644. - Philippe Deléham, Dec 12 2009

Cf. A098473, A007317, A090344, A000108, A000984, A007854, A076035, A076036, A106566.

Cf. A011973, A267633.

Sequence in context: A091187 A259824 A065173 * A153199 A056860 A158825

Adjacent sequences:  A098471 A098472 A098473 * A098475 A098476 A098477

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Sep 09 2004

EXTENSIONS

New name using a formula of Paul Barry by Peter Luschny, Feb 05 2015

STATUS

approved

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Last modified February 19 16:28 EST 2018. Contains 299356 sequences. (Running on oeis4.)