cp's OEIS Frontend

In general, for k>=1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 + (k-1)*r) / ((k+1)^(1/k) * sqrt(2*k*(k+1)*Pi*(1-r)) * n^(3/2) * r^(n + 1/k)), where r is the smallest real root of the equation (k+1)^(k+1) * r = k^k * (1-r)^k. - Vaclav Kotesovec, Nov 14 2021

Also, Riordan array (G,G), G(t)=(1 - ((1-5*t)/(1-t))^(1/2))/(2*t).

From Emanuele Munarini, May 18 2011: (Start)

Row sums = A002212.

Diagonal sums = A190737.

Central coefficients = A190738. (End)

G.f. = x*c(x)*c(x*c(x)) where c(x) is the generating function of the Catalan numbers C(n). Thus a(n) is the Catalan transform of the sequence C(n-1). Reference for the definition of Catalan transform is the paper by Paul Barry. - Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007

A129442 is an essentially identical sequence. - R. J. Mathar, Jun 13 2008

From Peter Bala, Jan 27 2020: (Start)

This sequence is the main diagonal of the lower triangular array formed by putting the sequence [0, 1, 1, 2, 5, 14, 42, ...] of Catalan numbers (with 0 prepended) in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.

0

1 1

1 2 2

2 4 6 6

5 9 15 21 21

14 23 38 59 80 80

...

Cf. A307495.

Alternatively, the sequence can be obtained by multiplying the sequence of Catalan numbers by the array A106566. (End)

See reference for precise definition.

From Petros Hadjicostas, Jan 12 2019: (Start)

In Cyvin et al. (1992), sequence (N(m): m >= 1) = (A002212(m): m >= 1) is defined by eq. (1), p. 533. (We may let N(0) := A002212(0) = 1.)

Sequence (M(m): m >= 1) is defined by eq. (13), p. 534. We have M(2*m) = M(2*m-1) = A007317(m) for m >= 1.

Sequences (N(m): m >= 1) and (M(m): m >= 1) appear in Table 1, p. 533.

The current sequence is denoted by 1^Q_(4+n) (with n = 1,2,3,...). Thus, a(n+4) = 1^Q_(4+n) for n >= 1; i.e., a(m) = 1^Q_{m} for m >= 5. We have 1^Q_(4+n) = (1/2)*(3*N(n) + M(n)) for n >= 1. See eq. (33), p. 536.

Sequence (1^Q_(4+n): n >= 1) appears in Table II, p. 537.

We may use the many formulae in the documentations of sequences A002212 and A007317 in order to create complicated formulae and recurrence relations for (a(n): n >= 5). We omit the details.

The first g.f. below is a combination of the g.f. for sequence A002212 by John W. Layman in 2001 and the g.f. for sequence A007317 by Ira M. Gessel and Jang Soo Kim in 2010.

The second g.f. appears in eq. (A1), p. 1180, in Cyvin et al. (1994). It is algebraically equivalent to the first g.f.

(Apparently, the word "annealated" in Cyvin et al. (1992) is spelled "annelated" in Cyvin et al. (1994).)

(End)

Setting both offsets to zero, this is the Catalan transform of A007317. - R. J. Mathar, Jun 29 2009

a(n) is also the cumulated sizes of admissible cuts of general rooted trees of size n. - Antoine Genitrini, Mar 14 2013

a(n) is the moment of order 2*n of the sum of two position operators in the weakly monotonne Fock space - Anna Kula, May 09 2025

a(n) is the moment of order 2*n of the measure with the density defined by g(x) = 1/(4*Pi) * (sqrt(sqrt(100-16x^2)-x^2-10) - sqrt(4-x^2)) if |x|<=2, g(x) = 1/(4*Pi) * sqrt((-2x^2-2|x|sqrt(4-x^2)+20) if 2 <= |x| <= 5/2 and g(x)=0 otherwise - Anna Kula, May 09 2025

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)

Binomial transform of A005159.

From Petros Hadjicostas, Jan 13 2019: (Start)

This sequence appears in Table I, p. 533, in Cyvin et al. (1992) and Table I, p. 1174, in Cyvin et al. (1994).

In Cyvin et al. (1992), it is defined through eq. (22), p. 535. We have a(n) = Sum_{i=1..n-1} M(i)*M(n-i), where M(2*n) = M(2*n-1) = A007317(n) for n >= 1.

In Cyvin et al. (1992), it is used in the calculation of sequence A026118. See eq. (34), p. 536, in Cyvin et al. (1992).

(The word "annelated" in the title of Cyvin et al. (1994) is spelled "annealated" in the text of Cyvin et al. (1992).)

(End)

Binomial transform of the first differences of the Catalan numbers (see A000245). - Paul Barry, Feb 16 2006

Starting (1, 5, 21, ...) = A002212, (1, 3, 10, 36, 137, ...) convolved with A007317, (1, 2, 5, 15, 51, ...). - Gary W. Adamson, May 19 2009

From Petros Hadjicostas, Jan 15 2019: (Start)

In Cyvin et al. (1992), sequence (N(m): m >= 1) = (A002212(m): m >= 1) is defined by eq. (1), p. 533. (We may let N(0) := A002212(0) = 1.)

In the same reference, sequence (M(m): m >= 1) is defined by eq. (13), p. 534. We have M(2*m) = M(2*m-1) = A007317(m) for m >= 1.

In the same reference, the sequence (M'(m): m >= 3) is defined by eq. (26), p. 535; see also Cyvin et al. (1994, Monatshefte fur Chemie), eq. 5, p. 1329. We have M'(m) = Sum_{1 <= i <= floor((m-1)/2)} N(i)*M(m-2*i) for m >= 3.

It turns out that M'(m) = a(floor((m + 1)/2)) for m >= 3, where (a(n): n >= 1) is the current sequence.

If 1 + U(x) = Sum_{n >= 0} N(n)*x^n = Sum_{n >= 0} A002212(n)*x^n, then the g.f. of the sequence (M(m): m >= 1) is V(x) = x*(1-x)^(-1)*(1 + U(x^2)). See eqs. 3 and 4, p. 1329, in Cyvin et al. (1994, Monatshefte fur Chemie).

Eq. 6 in the latter reference (pp. 1329-1330) states that the g.f. of the sequence (M'(m): m >= 3) is U(x^2)*V(x) = U(x^2)*x*(1-x)^(-1)*(1 + U(x^2)).

Since M'(m) = a(floor((m + 1)/2)) for m >= 3, the latter g.f. also equals (1 + x)*A(x^2)/x, where A(x) = Sum_{n >= 1} a(n)*x^n is the g.f. of the current sequence (given below by Emeric Deutsch).

Equating the two forms of the g.f. of the (M'(m): m >= 3), we get that A(x) = x*U(x)*(1 + U(x))/(1-x), where 1 + U(x) is the g.f. of A002212 (with U(0) = 0).

The sequence (M'(m): m >= 3) = (a(floor((m + 1)/2)): m >= 3) is used in the calculation of A026298 (= numbers of polyhexes of the class PF2 with three catafusenes annelated to pyrene).

(End)