A157073 Number of integer sequences of length n+1 with sum zero and sum of absolute values 48.

2, 144, 5762, 161480, 3493730, 61651128, 919453346, 11883194148, 135595653690, 1385919151540, 12835654787802, 108738668285884, 849286949294602, 6156408373152940, 41657479594194090, 264432781857156298, 1581589562174104296, 8947669593793415178

Offset: 1

Author

R. H. Hardin, Feb 22 2009

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (49,-1176,18424,-211876,1906884,-13983816,85900584, -450978066,2054455634,-8217822536,29135916264,-92263734836,262596783764, -675248872536,1575580702584,-3348108992991,6499270398159,-11554258485616, 18851684897584,-28277527346376,39049918716424,-49699896548176,58343356817424, -63205303218876,63205303218876,-58343356817424,49699896548176,-39049918716424, 28277527346376,-18851684897584,11554258485616,-6499270398159,3348108992991, -1575580702584,675248872536,-262596783764,92263734836,-29135916264,8217822536, -2054455634,450978066,-85900584,13983816,-1906884,211876,-18424,1176,-49,1).

Crossrefs

Cf. A103881, A156554.

Programs

Mathematica

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
A157073[n_]:= A103881[n, 24];
Table[A157073[n], {n, 50}] (* G. C. Greubel, Jan 27 2022 *)

Sage

def A103881(n,k): return sum( binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k) for i in (0..n) )
def A157073(n): return A103881(n, 24)
[A157073(n) for n in (1..50)] # G. C. Greubel, Jan 27 2022

Formula

a(n) = T(n,24); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).

From G. C. Greubel, Jan 27 2022: (Start)

a(n) = (n+1)*binomial(n+23, 24)*Hypergeometric3F2([-23, -n, 1-n], [2, -n-23], 1).

a(n) = (32247603683100/48!)*n*(n+1)*(16039842467222136539155708423040296550400000000 + 44525931866763275880171946837357992345600000000*n + 80162352992638760747141669078132808744960000000*n^2 + 83332132056036918488105323040316226063564800000*n^3 + 73898939901046923323215546964133115613675520000*n^4 + 44723032603767485653970945505703213072908288000*n^5 + 25612689570363639698514348299721610493952000000*n^6 + 10480812936564898576921267191518638010904084480*n^7 + 4344005319097142489606724072829615182825652224*n^8 + 1297122051885262240041808754289430257096523776*n^9 + 414887762782195453530600601421093882956775424*n^10 + 94644812314641495323136291475493075984289792*n^11 + 24355352682168634128406213057069994741673984*n^12 + 4374473519129303099715556660819067718420480*n^13 + 932704708306541825734118078032140866985984*n^14 + 134664684009917015892204293368196583403264*n^15 + 24318248584827829951503296783169426424064*n^16 + 2863809547176445630879170275831524932864*n^17 + 445554853840168519046977908135462996864*n^18 + 43232734952768495830917555723936691056*n^19 + 5874830266761134611938171223806383184*n^20 + 472840057219714797927879342154953928*n^21 + 56755099941609678328578532372768784*n^22 + 3803612022719773196434862241794913*n^23 + 407080783477921724014741379761599*n^24 + 22745052288898786827887020700757*n^25 + 2187954196667457627798376601499*n^26 + 101789002477485622214691512935*n^27 + 8861565620717173319451105401*n^28 + 341896269373157379303910179*n^29 + 27099899470126559155285701*n^30 + 860938389633999087289098*n^31 + 62459741766357695776566*n^32 + 1615864725980444668850*n^33 + 107800168679533475566*n^34 + 2233886413294116126*n^35 + 137618394169017186*n^36 + 2229052716036366*n^37 + 127282327855386*n^38 + 1552111826309*n^39 + 82428676891*n^40 + 711564777*n^41 + 35254791 n^42 + 192027*n^43 + 8901*n^44 + 23*n^45 + n^46).

G.f.: 2*x*(1 + 23*x + 529*x^2 + 5819*x^3 + 64009*x^4 + 448063*x^5 + 3136441*x^6 + 15682205*x^7 + 78411025*x^8 + 297961895*x^9 + 1132255201*x^10 + 3396765603*x^11 + 10190296809*x^12 + 24747863679*x^13 + 60101954649*x^14 + 120203909298*x^15 + 240407818596*x^16 + 400679697660*x^17 + 667799496100*x^18 + 934919294540*x^19 + 1308887012356*x^20 + 1546866469148*x^21 + 1828114918084*x^22 + 1828114918084*x^23 + 1828114918084*x^24 + 1546866469148*x^25 + 1308887012356*x^26 + 934919294540*x^27 + 667799496100*x^28 + 400679697660*x^29 + 240407818596*x^30 + 120203909298*x^31 + 60101954649*x^32 + 24747863679*x^33 + 10190296809*x^34 + 3396765603*x^35 + 1132255201*x^36 + 297961895*x^37 + 78411025*x^38 + 15682205*x^39 + 3136441*x^40 + 448063*x^41 + 64009*x^42 + 5819*x^43 + 529*x^44 + 23*x^45 + x^46)/(1-x)^49. (End)