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A157067
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Number of integer sequences of length n+1 with sum zero and sum of absolute values 36.
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1
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2, 108, 3242, 68190, 1107920, 14692734, 164826956, 1604095524, 13799638910, 106481351240, 745616925614, 4783532975546, 28342922553764, 156153427053890, 804648531335960, 3897769097766104, 17828728267167326, 77310179609631564, 318931533062574470
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OFFSET
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1,1
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LINKS
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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).
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FORMULA
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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).
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)
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MATHEMATICA
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A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n, -n, 1-k}, {2, 1-n - k}, 1];
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PROG
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(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) )
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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