login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A244129 Triangle read by rows: terms of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k). 28
0, 1, 0, 2, -2, 0, 3, -12, 9, 0, 4, -48, 108, -64, 0, 5, -160, 810, -1280, 625, 0, 6, -480, 4860, -15360, 18750, -7776, 0, 7, -1344, 25515, -143360, 328125, -326592, 117649, 0, 8, -3584, 122472, -1146880, 4375000, -7838208, 6588344, -2097152 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

T(n,k) = (-k)^(k-1) * k^(n-k) * binomial(n,k) for k>0, while T(n,0)=0 by convention.

LINKS

Stanislav Sykora, Table of n, a(n) for rows 1..100

S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(11), with b=1.

FORMULA

E.g.f. A(x,y) satisfies: A(x,y) * exp( A(x,y) ) = y*x*exp(x). - Paul D. Hanna, Sep 13 2017

EXAMPLE

First rows of the triangle, starting at row n=1. All rows sum up to 0, except the first one whose sum is 1:

0, 1;

0, 2, -2;

0, 3, -12, 9;

0, 4, -48, 108, -64;

0, 5, -160, 810, -1280, 625;

0, 6, -480, 4860, -15360, 18750, -7776;

0, 7, -1344, 25515, -143360, 328125, -326592, 117649;

0, 8, -3584, 122472, -1146880, 4375000, -7838208, 6588344, -2097152; ...

From Paul D. Hanna, Sep 13 2017: (Start)

E.g.f.: A(x,y) = y*x + (-2*y^2 + 2*y)*x^2/2! + (9*y^3 - 12*y^2 + 3*y)*x^3/3! + (-64*y^4 + 108*y^3 - 48*y^2 + 4*y)*x^4/4! + (625*y^5 - 1280*y^4 + 810*y^3 - 160*y^2 + 5*y)*x^5/5! + (-7776*y^6 + 18750*y^5 - 15360*y^4 + 4860*y^3 - 480*y^2 + 6*y)*x^6/6! + (117649*y^7 - 326592*y^6 + 328125*y^5 -  143360*y^4 + 25515*y^3 - 1344*y^2 + 7*y)*x^7/7! +...

such that A(x,y) * exp( A(x,y) ) = y*x*exp(x). (End)

PROG

(PARI) seq(nmax, b)={my(v, n, k, irow);

v = vector((nmax+1)*(nmax+2)/2-1);

for(n=1, nmax, irow=n*(n+1)/2; v[irow]=0;

  for(k=1, n, v[irow+k]=(-1)^(k-1)*(k*b)^(n-1)*binomial(n, k); ); );

return(v); }

a=seq(100, 1);

CROSSREFS

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A336978 A011137 A143396 * A342987 A090657 A167001

Adjacent sequences:  A244126 A244127 A244128 * A244130 A244131 A244132

KEYWORD

sign,tabl

AUTHOR

Stanislav Sykora, Jun 22 2014

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 April 11 16:13 EDT 2021. Contains 342886 sequences. (Running on oeis4.)