A103881 -id:A103881 - cp's OEIS frontend

A157060 Number of integer sequences of length n+1 with sum zero and sum of absolute values 22.

2, 66, 1212, 15620, 155850, 1272810, 8823080, 53265960, 285510150, 1379301990, 6078578508, 24680519604, 93093230958, 328512273390, 1091144804400, 3429182092560, 10244035242630, 29206656395910, 79759293448100

Offset: 1

R. H. Hardin, Feb 22 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (23,-253,1771,-8855,33649,-100947,245157,-490314, 817190,-1144066,1352078, -1352078,1144066,-817190,490314,-245157,100947,-33649, 8855,-1771,253,-23,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2,1-n - k}, 1];
A157060[n_] := A103881[n, 11];
Table[A157060[n], {n, 50}] (* G. C. Greubel, Jan 24 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 A157060(n): return A103881(n, 11)
[A157060(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,11); 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 24 2022: (Start)
a(n) = (n+1)*binomial(n+10, 11)*Hypergeometric3F2([-10, -n, 1-n], [2, -n-10], 1).
a(n) = (705432/22!)*n*(n+1)*(144850083840000 +292579402752000*n +440986525516800*n^2 +325146872079360*n^3 +235868591146176*n^4 +94960596391200*n^5 +43658519177360*n^6 +10953312870160*n^7 +3585704220196*n^8 +593523073650*n^9 +147783744195*n^10 +16467776610*n^11 +3255909581*n^12 +242376100*n^13 +39230830*n^14 +1873860*n^15 +254046*n^16 +7050*n^17 +815*n^18 +10*n^19 +n^20).
G.f.: 2*x*(1 +10*x +100*x^2 +450*x^3 +2025*x^4 +5400*x^5 +14400*x^6 +25200*x^7 +44100*x^8 +52920*x^9 +63504*x^10 +52920*x^11 +44100*x^12 +25200*x^13 +14400*x^14 +5400*x^15 +2025*x^16 +450*x^17 +100*x^18 +10*x^19 +x^20)/(1-x)^23. (End)

A157061 Number of integer sequences of length n+1 with sum zero and sum of absolute values 24.

Original entry on oeis.org

2, 72, 1442, 20260, 220250, 1958460, 14768810, 96900810, 563873400, 2953859370, 14097919968, 61908797418, 252208679268, 959882556570, 3433533723900, 11603837100660, 37221177046410, 113779617228060, 332648955112250, 933146517188760, 2518877938240202

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 (25,-300,2300,-12650,53130,-177100,480700,-1081575, 2042975,-3268760,4457400,-5200300,5200300,-4457400,3268760,-2042975,1081575, -480700,177100,-53130,12650,-2300,300,-25,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2,1-n - k}, 1];
A157061[n_]:= A103881[n, 12];
Table[A157061[n], {n, 50}] (* G. C. Greubel, Jan 24 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 A157061(n): return A103881(n, 12)
[A157061(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,12); 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 24 2022: (Start)
a(n) = (n+1)*binomial(n+11, 12)*Hypergeometric3F2([-11, -n, 1-n], [2, -n-11], 1).
a(n) = (2704156/24!)*n*(n+1)*(19120211066880000 + 40213832085504000*n + 61866024285081600*n^2 + 47770238895160320*n^3 + 35477403021764352*n^4 + 15129353226246336*n^5 + 7138320279252096*n^6 + 1926081009812080*n^7 + 648411230685152*n^8 + 117787792143956*n^9 + 30215435337736*n^10 + 3799367698665*n^11 + 775177128207*n^12 + 67808650591*n^13 + 11342892341*n^14 + 678888650*n^15 + 95251222*n^16 + 3725106*n^17 + 446226*n^18 + 10285*n^19 + 1067*n^20 + 11*n^21 + n^22).
G.f.: 2*x*(1 + 11*x + 121*x^2 + 605*x^3 + 3025*x^4 + 9075*x^5 + 27225*x^6 + 54450*x^7 + 108900*x^8 + 152460*x^9 + 213444*x^10 + 213444*x^11 + 213444*x^12 + 152460*x^13 + 108900*x^14 + 54450*x^15 + 27225*x^16 + 9075*x^17 + 3025*x^18 + 605*x^19 + 121*x^20 + 11*x^21 + x^22)/(1-x)^25. (End)

A157062 Number of integer sequences of length n+1 with sum zero and sum of absolute values 26.

Original entry on oeis.org

2, 78, 1692, 25740, 302850, 2912910, 23744840, 168278760, 1056789450, 5968878630, 30684132468, 144977296932, 634756203018, 2593322651430, 9946019437200, 35995371261360, 123490242018990, 403237594259010, 1257743358034100, 3759426449644740, 10799525727846702

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 (27,-351,2925,-17550,80730,-296010,888030,-2220075, 4686825,-8436285,13037895,-17383860,20058300,-20058300,17383860,-13037895, 8436285,-4686825,2220075,-888030,296010,-80730,17550,-2925,351,-27,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
A157062[n_]:= A103881[n, 13];
Table[A157062[n], {n, 50}] (* G. C. Greubel, Jan 24 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 A157062(n): return A103881(n, 13)
[A157062(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,13); 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 24 2022: (Start)
a(n) = (n+1)*binomial(n+12, 13)*Hypergeometric3F2([-12, -n, 1-n], [2, -n-12], 1).
a(n) = (10400600/26!)*n*(n+1)*(2982752926433280000 + 6502800338141184000*n + 10192999816651161600*n^2 + 8194549559065989120*n^3 + 6217354001317404672*n^4 + 2785907939555600640*n^5 + 1345736958526293696*n^6 + 386128480881709632*n^7 + 133329525393692848*n^8 + 26155830342678960*n^9 + 6893260441243396*n^10 + 955286585044572*n^11 + 200534847420673*n^12 + 19880275030680*n^13 + 3426180791086*n^14 + 242021337492*n^15 + 35027635423*n^16 + 1724131200*n^17 + 213288856*n^18 + 6959172*n^19 + 746383*n^20 + 14520*n^21 + 1366*n^22 + 12*n^23 + n^24).
G.f.: 2*x*(1 + 12*x + 144*x^2 + 792*x^3 + 4356*x^4 + 14520*x^5 + 48400*x^6 + 108900*x^7 + 245025*x^8 + 392040*x^9 + 627264*x^10 + 731808*x^11 + 853776*x^12 + 731808*x^13 + 627264*x^14 + 392040*x^15 + 245025*x^16 + 108900*x^17 + 48400*x^18 + 14520*x^19 + 4356*x^20 + 792*x^21 + 144*x^22 + 12*x^23 + x^24)/(1-x)^27. (End)

A157063 Number of integer sequences of length n+1 with sum zero and sum of absolute values 28.

Original entry on oeis.org

2, 84, 1962, 32130, 406800, 4208610, 36881420, 280819260, 1893408750, 11472968760, 63221641758, 319917948246, 1498750896708, 6545498596110, 26808012135000, 103501142484360, 378407481456870, 1315394383751460, 4363052456797550, 13853429338548630

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 (29, -406, 3654, -23751, 118755, -475020, 1560780, -4292145, 10015005, -20030010, 34597290, -51895935, 67863915, -77558760, 77558760, -67863915, 51895935, -34597290, 20030010, -10015005, 4292145, -1560780, 475020, -118755, 23751, -3654, 406, -29, 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];
A157063[n_]:= A103881[n, 14];
Table[A157063[n], {n, 50}] (* G. C. Greubel, Jan 24 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 A157063(n): return A103881(n, 14)
[A157063(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,14); 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 24 2022: (Start)
a(n) = (n+1)*binomial(n+13, 14)*Hypergeometric3F2([-13, -n, 1-n], [2, -n-13], 1).
a(n) = (40116600/28!)*n*(n+1)*(542861032610856960000 + 1222285449585328128000*n + 1949147924290921267200*n^2 + 1623917017366475120640*n^3 + 1256475121883342659584*n^4 + 587860847016245577216*n^5 + 290144191606881266304*n^6 + 87769963981312971072*n^7 + 31017509312522326880*n^8 + 6493644952485577744*n^9 + 1754084550497496360*n^10 + 263474544214276252*n^11 + 56764614862429890*n^12 + 6225163072052509*n^13 + 1102423601827845*n^14 + 88588233707662*n^15 + 13189509162960*n^16 + 769151138899*n^17 + 97984044015*n^18 + 4039324432*n^19 + 446558970*n^20 + 12345619*n^21 + 1198275*n^22 + 19942*n^23 + 1716*n^24 + 13*n^25 + n^26).
G.f.: 2*x*(1 + 13*x + 169*x^2 + 1014*x^3 + 6084*x^4 + 22308*x^5 + 81796*x^6 + 204490*x^7 + 511225*x^8 + 920205*x^9 + 1656369*x^10 + 2208492*x^11 + 2944656*x^12 + 2944656*x^13 + 2944656*x^14 + 2208492*x^15 + 1656369*x^16 + 920205*x^17 + 511225*x^18 + 204490*x^19 + 81796*x^20 + 22308*x^21 + 6084*x^22 + 1014*x^23 + 169*x^24 + 13*x^25 + x^26)/(1-x)^29. (End)

A157064 Number of integer sequences of length n+1 with sum zero and sum of absolute values 30.

Original entry on oeis.org

2, 90, 2252, 39500, 535502, 5930022, 55599992, 452715672, 3262336002, 21114177018, 124188986196, 670283877588, 3346707628446, 15564971674518, 67830161708592, 278406848295312, 1081149205136382, 3988232552194662, 14025412751733092, 47171740235162340

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 (31,-465,4495,-31465,169911,-736281,2629575, -7888725,20160075,-44352165,84672315,-141120525,206253075,-265182525,300540195, -300540195,265182525,-206253075,141120525,-84672315,44352165,-20160075,7888725, -2629575,736281,-169911,31465,-4495,465,-31,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
A157064[n_]:= A103881[n, 15];
Table[A157064[n], {n, 50}] (* G. C. Greubel, Jan 24 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 A157064(n): return A103881(n, 15)
[A157064(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,15); 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 24 2022: (Start)
a(n) = (n+1)*binomial(n+14, 15)*Hypergeometric3F2([-14, -n, 1-n], [2, -n-14], 1).
a(n) = (155117520/30!)*n*(n+1)*(114000816848279961600000 + 264279998869470904320000*n + 428198206877484244992000*n^2 + 368310644587032673075200*n^3 + 290167678780290006589440*n^4 + 141041429579778368449536*n^5 + 71004668064572092241664*n^6 + 22493711118572061653376*n^7 + 8120370606956264477184*n^8 + 1797910570397283902560*n^9 + 496779939204280228640*n^10 + 79886837991962961960*n^11 + 17626771834821917040*n^12 + 2101988853205045350*n^13 + 381651017327064975*n^14 + 34037459504198850*n^15 + 5201044031664375*n^16 + 346174867450230*n^17 + 45303425489595*n^18 + 2220034746930*n^19 + 252351294195*n^20 + 8844405570*n^21 + 883381005*n^22 + 20963670*n^23 + 1857765*n^24 + 26754*n^25 + 2121*n^26 + 14*n^27 + n^28).
G.f.: 2*x*(1 + 14*x + 196*x^2 + 1274*x^3 + 8281*x^4 + 33124*x^5 + 132496*x^6 + 364364*x^7 + 1002001*x^8 + 2004002*x^9 + 4008004*x^10 + 6012006*x^11 + 9018009*x^12 + 10306296*x^13 + 11778624*x^14 + 10306296*x^15 + 9018009*x^16 + 6012006*x^17 + 4008004*x^18 + 2004002 x^19 + 1002001*x^20 + 364364*x^21 + 132496*x^22 + 33124*x^23 + 8281*x^24 + 1274*x^25 + 196*x^26 + 14*x^27 + x^28)/(1-x)^31. (End)

A157065 Number of integer sequences of length n+1 with sum zero and sum of absolute values 32.

Original entry on oeis.org

2, 96, 2562, 47920, 692610, 8174544, 81659522, 708113304, 5431848930, 37403270520, 233931828834, 1341750437352, 7114703302434, 35117045235720, 162298598439330, 705951252118284, 2903050518427962, 11331495633292524, 42132555868774010, 149703679118108220

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 (33,-528,5456,-40920,237336,-1107568,4272048, -13884156,38567100,-92561040,193536720,-354817320,573166440,-818809200, 1037158320,-1166803110,1166803110,-1037158320,818809200,-573166440,354817320, -193536720,92561040,-38567100,13884156,-4272048,1107568,-237336,40920,-5456, 528,-33,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
A157065[n_]:= A103881[n, 16];
Table[A157065[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 A157065(n): return A103881(n, 16)
[A157065(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

a(n) = T(n,16); 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+15, 16)*Hypergeometric3F2([-15, -n, 1-n], [2, -n-15], 1).
a(n) = (601080390/32!)*n*(n+1)*(27360196043587190784000000 + 65137211981397216460800000*n + 107110050449356033228800000*n^2 + 94817527804050105212928000*n^3 + 75961411427539608595660800*n^4 + 38202458280851158526730240*n^5 + 19587950887554046039781376*n^6 + 6463560689425876180435200*n^7 + 2379792991631228553219840*n^8 + 553304095999692103772160*n^9 + 156114125142340061791744*n^10 + 26624540206135314300000*n^11 + 6005394587432947709600*n^12 + 768878902291539639600*n^13 + 142854837644598236640*n^14 + 13893755540913698625*n^15 + 2174500936993696575*n^16 + 161097628663020825*n^17 + 21612664028370855*n^18 + 1212359721607125*n^19 + 141388292047275*n^20 + 5907926749725*n^21 + 605873224515*n^22 + 18281995875*n^23 + 1664663325*n^24 + 34287435*n^25 + 2794869*n^26 + 35175*n^27 + 2585*n^28 + 15*n^29 + n^30).
G.f.: 2*x*(1 + 15*x + 225*x^2 + 1575*x^3 + 11025*x^4 + 47775*x^5 + 207025*x^6 + 621075*x^7 + 1863225*x^8 + 4099095*x^9 + 9018009*x^10 + 15030015*x^11 + 25050025*x^12 + 32207175*x^13 + 41409225*x^14 + 41409225*x^15 + 41409225*x^16 + 32207175*x^17 + 25050025*x^18 + 15030015*x^19 + 9018009*x^20 + 4099095*x^21 + 1863225*x^22 + 621075*x^23 + 207025*x^24 + 47775*x^25 + 11025*x^26 + 1575*x^27 + 225*x^28 + 15*x^29 + x^30)/(1-x)^33. (End)

A157066 Number of integer sequences of length n+1 with sum zero and sum of absolute values 34.

Original entry on oeis.org

2, 102, 2892, 57460, 882030, 11053434, 117206264, 1078467624, 8774904690, 64062783510, 424600608564, 2579499722124, 14479567043214, 75613799423610, 369504358622640, 1698353774374704, 7375213677918294, 30379740299612514, 119122913376492980, 446056011713374860

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 (35,-595,6545,-52360,324632,-1623160,6724520, -23535820,70607460,-183579396,417225900,-834451800,1476337800,-2319959400, 3247943160,-4059928950,4537567650,-4537567650,4059928950,-3247943160,2319959400, -1476337800,834451800,-417225900,183579396,-70607460,23535820,-6724520,1623160, -324632,52360,-6545,595,-35,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
A157066[n_]:= A103881[n, 17];
Table[A157066[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 A157066(n): return A103881(n, 17)
[A157066(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

a(n) = T(n,17); 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+16, 17)*Hypergeometric3F2([-16, -n, 1-n], [2, -n-16], 1).
a(n) = (2333606220/34!)*n*(n+1)*(7441973323855715893248000000 + 18155084795637437929881600000*n + 30268626521952180908851200000*n^2 + 27504128369891325149577216000*n^3 + 22380511931408981359868313600*n^4 + 11606451235232148856801198080*n^5 + 6053325843616709826370609152*n^6 + 2071495721724703057714876416*n^7 + 776772176331488107582976256*n^8 + 188575401978015909077544960*n^9 + 54249004662342491124700928*n^10 + 9739700938346246478267904*n^11 + 2242198636428402181902944*n^12 + 305221374822225945324800*n^13 + 57932851765719841948880*n^14 + 6064778909442097812240*n^15 + 970512936702416581665*n^16 + 78610569988240809600*n^17 + 10791805239981923160*n^18 + 675564468731071680*n^19 + 80680394732550780*n^20 + 3869168748681600*n^21 + 406620563860680*n^22 + 14666674470240*n^23 + 1369455578790*n^24 + 35960795520*n^25 + 3007754088*n^26 + 54285504*n^27 + 4095964*n^28 + 45440*n^29 + 3112*n^30 + 16*n^31 + n^32).
G.f.: 2*x*(1 + 16*x + 256*x^2 + 1920*x^3 + 14400*x^4 + 67200*x^5 + 313600*x^6 + 1019200*x^7 + 3312400*x^8 + 7949760*x^9 + 19079424*x^10 + 34978944*x^11 + 64128064*x^12 + 91611520*x^13 + 130873600*x^14 + 147232800*x^15 + 165636900*x^16 + 147232800*x^17 + 130873600*x^18 + 91611520*x^19 + 64128064*x^20 + 34978944*x^21 + 19079424*x^22 + 7949760*x^23 + 3312400*x^24 + 1019200*x^25 + 313600*x^26 + 67200*x^27 + 14400*x^28 + 1920*x^29 + 256*x^30 + 16*x^31 + x^32)/(1-x)^35. (End)

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

Original entry on oeis.org

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)

cp's OEIS Frontend

A157060 Number of integer sequences of length n+1 with sum zero and sum of absolute values 22.

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A157061 Number of integer sequences of length n+1 with sum zero and sum of absolute values 24.

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A157062 Number of integer sequences of length n+1 with sum zero and sum of absolute values 26.

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A157063 Number of integer sequences of length n+1 with sum zero and sum of absolute values 28.

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A157064 Number of integer sequences of length n+1 with sum zero and sum of absolute values 30.

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A157065 Number of integer sequences of length n+1 with sum zero and sum of absolute values 32.

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A157066 Number of integer sequences of length n+1 with sum zero and sum of absolute values 34.

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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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