A157072 Number of integer sequences of length n+1 with sum zero and sum of absolute values 46.

2, 138, 5292, 142140, 2947590, 49858158, 712832792, 8832976488, 96648771870, 947399938870, 8416542780492, 68407265558268, 512700872216442, 3567168162771570, 23172711963346320, 141251698411654288, 811481822951916942, 4410812923746903558, 22762369531189431140

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 (47,-1081,16215,-178365,1533939,-10737573,62891499, -314457495,1362649145,-5178066751,17417133617,-52251400851,140676848445, -341643774795,751616304549,-1503232609098,2741188875414,-4568648125690, 6973199770790,-9762479679106,12551759587422,-14833897694226,16123801841550, -16123801841550,14833897694226,-12551759587422,9762479679106,-6973199770790, 4568648125690,-2741188875414,1503232609098,-751616304549,341643774795, -140676848445,52251400851,-17417133617,5178066751,-1362649145,314457495, -62891499,10737573,-1533939,178365,-16215,1081,-47,1).

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];
A157072[n_]:= A103881[n, 23];
Table[A157072[n], {n, 50}] (* G. C. Greubel, Jan 25 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 A157072(n): return A103881(n, 23)
[A157072(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

Formula

a(n) = T(n,23); 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 25 2022: (Start)

a(n) = (n+1)*binomial(n+22, 23)*Hypergeometric3F2([-22, -n, 1-n], [2, -n-22], 1).

a(n) = (8233430727600/46!)*n*(n+1)*(29057685629025609672383529751884595200000000 + 79452183147274795032078126183088128000000000*n + 141714570491802789957788787173889146880000000*n^2 + 145059233577401185360645255317602854502400000*n^3 + 127311238631698355225728753712566590504960000*n^4 + 75715351658040622253223159728830038933504000*n^5 + 42877191833222765234078376290791889436672000*n^6 + 17200430297827490899524392276866711148298240*n^7 + 7044053985717499896347935293286272148242432*n^8 + 2056356540242318373959793917651894923345920*n^9 + 649440492446852015686988427724931399725056*n^10 + 144397972805007063337564416010542851069952*n^11 + 36667320366669588030104490299079773399040*n^12 + 6396965852709968433012959028877233569280*n^13 + 1345127187454407600202359730144941101312*n^14 + 187910794743597175883242789084896626944*n^15 + 33447938991896902409607083541643054848*n^16 + 3794396649208001585975013323140823680*n^17 + 581596730556665903213714682678333648*n^18 + 54086974909357210248192242794085176*n^19 + 7237583584021550113709859989257256*n^20 + 555028323889889756001001018844270*n^21 + 65573979319258648679066391179799*n^22 + 4158352352131928037710752254818*n^23 + 437873818310682613098943721859*n^24 + 22960062441581678852556730250*n^25 + 2172171883621041163474766945*n^26 + 93893204989495788867340350*n^27 + 8036153654616364534710453*n^28 + 284537563980038034430380*n^29 + 22164572970995075714214*n^30 + 636147121922304974388*n^31 + 45339923676136414270*n^32 + 1038127683748744820*n^33 + 68016631509831858*n^34 + 1212869363347796*n^35 + 73356699164562*n^36 + 981609846470*n^37 + 55012667347*n^38 + 519602314*n^39 + 27075279*n^40 + 160930*n^41 + 7821*n^42 + 22*n^43 + n^44).

G.f.: 2*x*(1 + 22*x + 484*x^2 + 5082*x^3 + 53361*x^4 + 355740*x^5 + 2371600*x^6 + 11265100*x^7 + 53509225*x^8 + 192633210*x^9 + 693479556*x^10 + 1964858742*x^11 + 5567099769*x^12 + 12724799472*x^13 + 29085255936*x^14 + 54534854880*x^15 + 102252852900*x^16 + 159059993400*x^17 + 247426656400*x^18 + 321654653320*x^19 + 418151049316*x^20 + 456164781072*x^21 + 497634306624*x^22 + 456164781072*x^23 + 418151049316*x^24 + 321654653320*x^25 + 247426656400*x^26 + 159059993400*x^27 + 102252852900*x^28 + 54534854880*x^29 + 29085255936*x^30 + 12724799472*x^31 + 5567099769*x^32 + 1964858742*x^33 + 693479556*x^34 + 192633210*x^35 + 53509225*x^36 + 11265100*x^37 + 2371600*x^38 + 355740*x^39 + 53361*x^40 + 5082*x^41 + 484*x^42 + 22*x^43 + x^44)/(1-x)^47. (End)