A103881 -id:A103881 - cp's OEIS frontend

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

R. H. Hardin, Feb 22 2009

nonn

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).

Cf. A103881, A156554.

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 *)

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

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)

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

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Feb 22 2009

nonn

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).

Cf. A103881, A156554.

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 *)

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

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)

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

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

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A177326 Number of permutations of n copies of 1..5 with all adjacent differences <= 3 in absolute value.

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