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

1,1

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).

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)

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 *)

Prog

(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

Crossrefs

Cf. A103881, A156554.

Sequence in context: A195379 A221227 A087619 * A051029 A212715 A329594

Adjacent sequences: A157069 A157070 A157071 * A157073 A157074 A157075

Keyword

nonn

Author

R. H. Hardin, Feb 22 2009

Status

approved