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

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A304638 a(n) equals the coefficient of x^n in Sum_{m>=0} (x^m + 1/x^m)^m for n > 0. 8
1, 0, 3, 1, 10, 0, 35, 4, 127, 0, 462, 15, 1716, 0, 6440, 57, 24310, 0, 92378, 210, 352737, 0, 1352078, 798, 5200301, 0, 20058384, 3003, 77558760, 0, 300540195, 11468, 1166803440, 0, 4537567657, 43759, 17672631900, 0, 68923265697, 168080, 269128937220, 0, 1052049481860, 646646, 4116715368841, 0, 16123801841550, 2496647, 63205303218877, 0, 247959266493500, 9657700, 973469712824056, 0, 3824345300380385, 37444162, 15033633249846102, 0, 59132290782430712, 145422720, 232714176627630544, 0, 916312070471589206, 565730729, 3609714217008133585, 0, 14226520737620288370, 2203961430, 56093138908332566782, 0, 221256270138418389602, 8597528644, 873065282167813104916, 0, 3446310324346635137703, 33578000610, 13608507434599516007855, 0, 53753604366668088230810, 131282534380 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The coefficient of 1/x^n in Sum_{m>=0} (x^m + 1/x^m)^m equals a(n) for n > 0, while the constant term in the sum increases without limit.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..300

FORMULA

a(4*n + 2) = 0 for n >= 0.

a(n) = [x^n] Sum_{m>=0} x^(m^2) * (1 + 1/x^(2*m))^m, for n > 0.

EXAMPLE

G.f.: A(x) = x + 3*x^3 + x^4 + 10*x^5 + 35*x^7 + 4*x^8 + 127*x^9 + 462*x^11 + 15*x^12 + 1716*x^13 + 6440*x^15 + 57*x^16 + 24310*x^17 + 92378*x^19 + 210*x^20 + 352737*x^21 + 1352078*x^23 + 798*x^24 + 5200301*x^25 + ...

RELATED SERIES.

The odd bisection of the g.f. begins:

(A(x) - A(-x))/2 = x + 3*x^3 + 10*x^5 + 35*x^7 + 127*x^9 + 462*x^11 + 1716*x^13 + 6440*x^15 + 24310*x^17 + 92378*x^19 + 352737*x^21 + 1352078*x^23 + 5200301*x^25 + 20058384*x^27 + 77558760*x^29 + 300540195*x^31 + 1166803440*x^33 + 4537567657*x^35 + 17672631900*x^37 + 68923265697*x^39 + 269128937220*x^41 + 1052049481860*x^43 + 4116715368841*x^45 + 16123801841550*x^47 + 63205303218877*x^49 + ... + A316596(n)*x^(2*n-1) + ...

The even bisection of the g.f. begins:

(A(x) + A(-x))/2 = x^4 + 4*x^8 + 15*x^12 + 57*x^16 + 210*x^20 + 798*x^24 + 3003*x^28 + 11468*x^32 + 43759*x^36 + 168080*x^40 + 646646*x^44 + 2496647*x^48 + 9657700*x^52 + 37444162*x^56 + 145422720*x^60 + 565730729*x^64 + 2203961430*x^68 + 8597528644*x^72 + 33578000610*x^76 + 131282534380*x^80 + ... + A316592(n)*x^(4*n) + ...

PROG

(PARI) {a(n) = polcoeff( sum(m=1, n, (x^-m + x^m)^m +x*O(x^n)), n, x)}

for(n=1, 80, print1(a(n), ", "))

CROSSREFS

Cf. A316590, A316591, A316592, A316593, A316594, A316595, A316596 (bisection).

Sequence in context: A227795 A090479 A227758 * A141903 A010289 A226646

Adjacent sequences:  A304635 A304636 A304637 * A304639 A304640 A304641

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 15 2018

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 December 5 10:25 EST 2020. Contains 338945 sequences. (Running on oeis4.)