login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A228899 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^(k+1) * y^k ), as read by rows. 1
1, 1, 1, 1, 3, 1, 1, 6, 12, 1, 1, 10, 71, 76, 1, 1, 15, 281, 2153, 701, 1, 1, 21, 861, 29166, 129509, 8477, 1, 1, 28, 2212, 244725, 7664343, 12391414, 126126, 1, 1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1, 1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Note that the following g.f. does NOT yield an integer triangle: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^k * y^k ).

LINKS

Table of n, a(n) for n=0..54.

EXAMPLE

This triangle begins:

1;

1, 1;

1, 3, 1;

1, 6, 12, 1;

1, 10, 71, 76, 1;

1, 15, 281, 2153, 701, 1;

1, 21, 861, 29166, 129509, 8477, 1;

1, 28, 2212, 244725, 7664343, 12391414, 126126, 1;

1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1;

1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1; ...

...

G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+12*y^2+y^3)*x^3 + (1+10*y+71*y^2+76*y^3+y^4)*x^4 + (1+15*y+281*y^2+2153*y^3+701*y^4+y^5)*x^5 +...

The logarithm of the g.f. equals the series:

log(A(x)) = (1 + x)*x

+ (1 + 2^2*x + x^2)*x^2/2

+ (1+ 3^2*y + 3^3*y^2 + y^3)*x^3/3

+ (1+ 4^2*y + 6^3*y^2 + 4^4*y^3 + x^4)*x^4/4

+ (1+ 5^2*y + 10^3*y^2 + 10^4*y^3 + 5^5*y^4 + y^5)*x^5/5

+ (1+ 6^2*y + 15^3*y^2 + 20^4*y^3 + 15^5*y^4 + 6^6*y^5 + y^6)*x^6/6 +...

in which the coefficients form A219207(n,k) = binomial(n, k)^(k+1).

PROG

(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m, j)^(j+1)*y^j))+x*O(x^n)), n, x), k, y)}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A184730 (row sums), A181070 (antidiagonal sums), A060946 (diagonal).

Cf. related triangles: A219207, A209424, A228904.

Sequence in context: A235116 A235114 A272866 * A255918 A102479 A228902

Adjacent sequences:  A228896 A228897 A228898 * A228900 A228901 A228902

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Sep 07 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 16 06:30 EDT 2019. Contains 328050 sequences. (Running on oeis4.)