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

1,1

Links

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).

Formula

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)

Mathematica

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 *)

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 A157067(n): return A103881(n, 18)
[A157067(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

Crossrefs

Cf. A103881, A156554.

Sequence in context: A224819 A156502 A111456 * A364616 A375841 A287153

Adjacent sequences: A157064 A157065 A157066 * A157068 A157069 A157070

Keyword

nonn

Author

R. H. Hardin, Feb 22 2009

Status

approved