A156554 -id:A156554 - cp's OEIS frontend

A157067 Number of integer sequences of length n+1 with sum zero and sum of absolute values 36.

2, 108, 3242, 68190, 1107920, 14692734, 164826956, 1604095524, 13799638910, 106481351240, 745616925614, 4783532975546, 28342922553764, 156153427053890, 804648531335960, 3897769097766104, 17828728267167326, 77310179609631564, 318931533062574470

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 (37,-666,7770,-66045,435897,-2324784,10295472, -38608020,124403620,-348330136,854992152,-1852482996,3562467300,-6107086800, 9364199760,-12875774670,15905368710,-17672631900,17672631900,-15905368710, 12875774670,-9364199760,6107086800,-3562467300,1852482996,-854992152,348330136, -124403620,38608020,-10295472,2324784,-435897,66045,-7770,666,-37,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
A157067[n_]:= A103881[n, 18];
Table[A157067[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 A157067(n): return A103881(n, 18)
[A157067(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

a(n) = T(n,18); 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+17, 18)*Hypergeometric3F2([-17, -n, 1-n], [2, -n-17], 1).
a(n) = (9075135300/36!)*n*(n+1)*(2277243837099849063333888000000 + 5681969493970603176728985600000*n + 9596433215362696956739584000000*n^2 + 8930829932059932571221098496000*n^3 + 7373779588191144329720945049600*n^4 + 3932042780814990233298927943680*n^5 + 2083614342312300867651696279552*n^6 + 736784230189243202709052538880*n^7 + 281032534792096725785629118976*n^8 + 70909200002908166006639354112*n^9 + 20771324838612576755137269504*n^10 + 3902581566393773771469894400*n^11 + 915676404299665995395824064*n^12 + 131515117514883976361738848*n^13 + 25463636023538740834106624*n^14 + 2840680826306519243676400*n^15 + 464075830766617076558690*n^16 + 40553554340342769625905*n^17 + 5687795599925219641425*n^18 + 390183416511400627800*n^19 + 47640166465301752080*n^20 + 2555532347549932860*n^21 + 274751324750187660*n^22 + 11400551973525000*n^23 + 1089674111434740*n^24 + 34284748268550*n^25 + 2937122649078*n^26 + 67743183720*n^27 + 5238258144*n^28 + 83536028*n^29 + 5866156*n^30 + 57800*n^31 + 3706*n^32 + 17*n^33 + n^34).
G.f.: 2*x*(1 + 17*x + 289*x^2 + 2312*x^3 + 18496*x^4 + 92480*x^5 + 462400*x^6 + 1618400*x^7 + 5664400*x^8 + 14727440*x^9 + 38291344*x^10 + 76582688*x^11 + 153165376*x^12 + 240688448*x^13 + 378224704*x^14 + 472780880*x^15 + 590976100*x^16 + 590976100*x^17 + 590976100*x^18 + 472780880*x^19 + 378224704*x^20 + 240688448*x^21 + 153165376*x^22 + 76582688*x^23 + 38291344*x^24 + 14727440*x^25 + 5664400*x^26 + 1618400*x^27 + 462400*x^28 + 92480*x^29 + 18496*x^30 + 2312*x^31 + 289*x^32 + 17*x^33 + x^34)/(1-x)^37. (End)

A157069 Number of integer sequences of length n+1 with sum zero and sum of absolute values 40.

Original entry on oeis.org

2, 120, 4002, 93500, 1687002, 24836196, 309182762, 3337508646, 31830097752, 272125000774, 2109875558208, 14977318285254, 98118326104708, 597217934730774, 3397036441760412, 18148572883826236, 91470993083858322, 436643312483178036, 1981038544180652162

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 (41, -820, 10660, -101270, 749398, -4496388, 22481940, -95548245, 350343565, -1121099408, 3159461968, -7898654920, 17620076360, -35240152720, 63432274896, -103077446706, 151584480450, -202112640600, 244662670200, -269128937220, 269128937220, -244662670200, 202112640600, -151584480450, 103077446706, -63432274896, 35240152720, -17620076360, 7898654920, -3159461968, 1121099408, -350343565, 95548245, -22481940, 4496388, -749398, 101270, -10660, 820, -41, 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];
A157069[n_]:= A103881[n, 20];
Table[A157069[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 A157069(n): return A103881(n, 20)
[A157069(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

a(n) = T(n,20); 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+19, 20)*Hypergeometric3F2([-19, -n, 1-n], [2, -n-19], 1).
a(n) = (137846528820/40!)*n*(n+1)*(295950609069496384270872084480000000 + 768802633735657375654446366720000000*n + 1329504929585504813849213140992000000*n^2 + 1290742342817244773843889039605760000*n^3 + 1094357439529328748458516078002176000*n^4 + 612766113778575140689735509285273600*n^5 + 334228753141512703020765378377809920*n^6 + 125103295909205358813292403873120256*n^7 + 49218727808847235410751174949228544*n^8 + 13269339361037181895414921845144576*n^9 + 4016584445427935868170163264804864*n^10 + 815165270428049073818572136963328*n^11 + 197974483136507211917478313071872*n^12 + 31108483670185057904409322050688*n^13 + 6244038933930696351877891958272*n^14 + 773683666573321735532607476256*n^15 + 131217385198850594964429765744*n^16 + 12969478215579974430537627276*n^17 + 1890935510804343168840278104*n^18 + 150029328423053669455781465*n^19 + 19066083072333125878657535*n^20 + 1216465853960978843551515*n^21 + 136285407600184771625385*n^22 + 6973959244303571061060*n^23 + 695382022718273834940*n^24 + 28325593615993410660*n^25 + 2534141220949541220*n^26 + 81059848291860174*n^27 + 6552284226337026*n^28 + 160984848978954*n^29 + 11828920639006*n^30 + 215437887572*n^31 + 14466923228*n^32 + 183962712*n^33 + 11343228*n^34 + 89889*n^35 + 5111*n^36 + 19*n^37 + n^38).
G.f.: 2*x*(1 + 19*x + 361*x^2 + 3249*x^3 + 29241*x^4 + 165699*x^5 + 938961*x^6 + 3755844*x^7 + 15023376*x^8 + 45070128*x^9 + 135210384*x^10 + 315490896*x^11 + 736145424*x^12 + 1367127216*x^13 + 2538950544*x^14 + 3808425816*x^15 + 5712638724*x^16 + 6982113996*x^17 + 8533694884*x^18 + 8533694884*x^19 + 8533694884*x^20 + 6982113996*x^21 + 5712638724*x^22 + 3808425816*x^23 + 2538950544*x^24 + 1367127216*x^25 + 736145424*x^26 + 315490896*x^27 + 135210384*x^28 + 45070128*x^29 + 15023376*x^30 + 3755844*x^31 + 938961*x^32 + 165699*x^33 + 29241*x^34 + 3249*x^35 + 361*x^36 + 19*x^37 + x^38)/(1-x)^41. (End)

A157070 Number of integer sequences of length n+1 with sum zero and sum of absolute values 42.

Original entry on oeis.org

2, 126, 4412, 108220, 2049770, 31674678, 413820584, 4687156248, 46894786710, 420487598410, 3418440803052, 25437258929836, 174630760523102, 1113521228343010, 6633053884912560, 37097838553993648, 195668363575483134, 977073310632294978, 4635352353992402420

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 (43,-903,12341,-123410,962598,-6096454,32224114, -145008513,563921995,-1917334783,5752004349,-15338678264,36576848168, -78378960360,151532656696,-265182149218,421171648758,-608359048206, 800472431850,-960566918220,1052049481860,-1052049481860,960566918220, -800472431850,608359048206,-421171648758,265182149218,-151532656696, 78378960360,-36576848168,15338678264,-5752004349,1917334783,-563921995, 145008513,-32224114,6096454,-962598,123410,-12341,903,-43,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
A157070[n_]:= A103881[n, 21];
Table[A157070[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 A157070(n): return A103881(n, 21)
[A157070(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

a(n) = T(n,21); 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+20, 21)*Hypergeometric3F2([-20, -n, 1-n], [2, -n-20], 1).
a(n) = (538257874440/42!)*n*(n+1)*(124299255809188481393766275481600000000 + 328816118350366025460284915712000000000*n + 574832876343430323089683765002240000000*n^2 + 568701882574952901291417659454259200000*n^3 + 488065218731065719417147635733626880000*n^4 + 279248916577588134058859235459858432000*n^5 + 154338522148314741971664420691673088000*n^6 + 59227959344696504761998117194266705920*n^7 + 23633263646950664110615399338389323776*n^8 + 6557497087812561104289290673945292800*n^9 + 2014840321470361119845933104915307520*n^10 + 422701102488339328367203562820695040*n^11 + 104284338041749995423701069631220992*n^12 + 17025052804207868558201481522726400*n^13 + 3473748992461285895698788698610560*n^14 + 449827918639409055961252979192960*n^15 + 77602697715487702683123150572128*n^16 + 8071528554520601160114398770800*n^17 + 1197769342263854188918636742220*n^18 + 100831028153769404548233777380*n^19 + 13049306298068383096447853569*n^20 + 892237320110273631864787000*n^21 + 101851737197591285675901050*n^22 + 5654771034611195278152900*n^23 + 574799001272234774582445*n^24 + 25804389773082709176000*n^25 + 2354558801452942771200*n^26 + 84727960701572097480*n^27 + 6988357410140155794*n^28 + 198659321097901200*n^29 + 14901112723277580*n^30 + 327062325560360*n^31 + 22429224033778*n^32 + 366602803600*n^33 + 23094295940*n^34 + 264617940*n^35 + 15377517*n^36 + 110200*n^37 + 5930*n^38 + 20*n^39 + n^40).
G.f.: 2*x*(1 + 20*x + 400*x^2 + 3800*x^3 + 36100*x^4 + 216600*x^5 + 1299600*x^6 + 5523300*x^7 + 23474025*x^8 + 75116880*x^9 + 240374016*x^10 + 600935040*x^11 + 1502337600*x^12 + 3004675200*x^13 + 6009350400*x^14 + 9765194400*x^15 + 15868440900*x^16 + 21157921200*x^17 + 28210561600*x^18 + 31031617760*x^19 + 34134779536*x^20 + 31031617760*x^21 + 28210561600*x^22 + 21157921200*x^23 + 15868440900*x^24 + 9765194400*x^25 + 6009350400*x^26 + 3004675200*x^27 + 1502337600*x^28 + 600935040*x^29 + 240374016*x^30 + 75116880*x^31 + 23474025*x^32 + 5523300*x^33 + 1299600*x^34 + 216600*x^35 + 36100*x^36 + 3800*x^37 + 400*x^38 + 20*x^39 + x^40)/(1-x)^43. (End)

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

Original entry on oeis.org

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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A157067 Number of integer sequences of length n+1 with sum zero and sum of absolute values 36.

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A157069 Number of integer sequences of length n+1 with sum zero and sum of absolute values 40.

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A157070 Number of integer sequences of length n+1 with sum zero and sum of absolute values 42.

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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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