A157071 Number of integer sequences of length n+1 with sum zero and sum of absolute values 44.

2, 132, 4842, 124410, 2468160, 39944058, 546468188, 6480461988, 67871267730, 636929126680, 5418075931842, 42175511270802, 302809501559292, 2018756213756730, 12569156781338280, 73453463921029288, 404681030359774038, 2110083485993094708, 10449172010467254110

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 (45,-990,14190,-148995,1221759,-8145060,45379620, -215553195,886163135,-3190187286,10150595910,-28760021745,73006209045, -166871334960,344867425584,-646626422970,1103068603890,-1715884494940, 2438362177020,-3169870830126,3773655750150,-4116715363800,4116715363800, -3773655750150,3169870830126,-2438362177020,1715884494940,-1103068603890, 646626422970,-344867425584,166871334960,-73006209045,28760021745,-10150595910, 3190187286,-886163135,215553195,-45379620,8145060,-1221759,148995,-14190,990, -45,1).

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];
A157071[n_]:= A103881[n, 22];
Table[A157071[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 A157071(n): return A103881(n, 22)
[A157071(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

Formula

a(n) = T(n,22); 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+21, 22)*Hypergeometric3F2([-21, -n, 1-n], [2, -n-21], 1).

a(n) = (2104098963720/44!)*n*(n+1)*(57426256183845078403920019272499200000000 + 154523331049862061871920722844057600000000*n + 272931042730182050308953933852180480000000*n^2 + 274811760152840277181518317732914790400000*n^3 + 238572405346744756522223079981931560960000*n^4 + 139262369052198098042953067132860760064000*n^5 + 77935938615959415459621013175403479040000*n^6 + 30604426182364394781448082978276352983040*n^7 + 12375921432622844665839937493301408202752*n^8 + 3525862191155367176504575677468901072896*n^9 + 1098754428098133845773800888636057839616*n^10 + 237599556100490353222412641226382566400*n^11 + 59493628751502324229674305656594696704*n^12 + 10055545494420785177786806925755237632*n^13 + 2083707534251437187516974041889280512*n^14 + 280769227253093987663773439057969280*n^15 + 49222409547083603766449249805178816*n^16 + 5358702844213759492318438394124288*n^17 + 808545762040841563486050098435368*n^18 + 71737091194582402868575376736180*n^19 + 9444831471354374079538688193458*n^20 + 686249074150382462946936518949*n^21 + 79734029925781576979739452669*n^22 + 4751390699139789217700561850*n^23 + 491813838546810047652354440*n^24 + 23992407101881141905543345*n^25 + 2230301742551313936748845*n^26 + 88590052674638107230960*n^27 + 7447195060372280892588*n^28 + 238536111960173626074*n^29 + 18243176552448224154*n^30 + 464026161597715500*n^31 + 32458785389210936*n^32 + 640384199972538*n^33 + 41164050437258*n^34 + 607233703020*n^35 + 36020303474*n^36 + 373840257*n^37 + 20541577*n^38 + 133770*n^39 + 6832*n^40 + 21*n^41 + n^42).

G.f.: 2*x*(1 + 21*x + 441*x^2 + 4410*x^3 + 44100*x^4 + 279300*x^5 + 1768900*x^6 + 7960050*x^7 + 35820225*x^8 + 121788765*x^9 + 414081801*x^10 + 1104218136*x^11 + 2944581696*x^12 + 6309817920*x^13 + 13521038400*x^14 + 23661817200*x^15 + 41408180100*x^16 + 59811815700*x^17 + 86394844900*x^18 + 103673813880*x^19 + 124408576656*x^20 + 124408576656*x^21 + 124408576656*x^22 + 103673813880*x^23 + 86394844900*x^24 + 59811815700*x^25 + 41408180100*x^26 + 23661817200*x^27 + 13521038400*x^28 + 6309817920*x^29 + 2944581696*x^30 + 1104218136*x^31 + 414081801*x^32 + 121788765*x^33 + 35820225*x^34 + 7960050*x^35 + 1768900*x^36 + 279300*x^37 + 44100*x^38 + 4410*x^39 + 441*x^40 + 21*x^41 + x^42)/(1-x)^45. (End)