Search: a364374 -id:a364374
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A364375
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G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^3).
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+10
3
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1, 0, -1, 2, 0, -11, 28, 1, -206, 564, 38, -4711, 13329, 1273, -119762, 344707, 41884, -3251250, 9445976, 1381154, -92305098, 269504686, 45848871, -2707126108, 7921304973, 1532928960, -81375728566, 238196143730, 51591751698, -2493907008116, 7293147604136
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0,4
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FORMULA
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a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1).
D-finite with recurrence +2*n*(191553133*n -462036810)*(2*n+1) *(n+1)*a(n) +2*n*(6735679202*n^3 -31340869996*n^2 +39568451245*n -13340358389)*a(n-1) +6*(13937077342*n^4 -106287464449*n^3 +278022830194*n^2 -296712736455*n +108876423952)*a(n-2) +6*(42118990776*n^4 -422141236704*n^3 +1546534911485*n^2 -2448212978721*n +1409411956166)*a(n-3) +6*(72631772298*n^4 -948761263665*n^3 +4512788370945*n^2 -9254886913710*n +6888712179986)*a(n-4) +3*(10147840245*n^4 -513806508936*n^3 +5519825354705*n^2 -22028093493130*n +30003008863784)*a(n-5) +6*(-9503341830*n^4 +235269814455*n^3 -2064338754902*n^2 +7709425316943*n -10409244067330)*a(n-6) +18*(3*n-20)*(n-6) *(156488131*n-746235854) *(3*n-13)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
G.f. A(x) satisfies (1/x) * series_reversion(x*A(x)) = 1/G(x), where G(x) is the g.f. of A364371.
P-recursive: fifth-order recurrence: (2*n+1)*(2*n+2)*(3045*n^5-26680*n^4+84901*n^3-123566*n^2+86300*n-25368)*n*a(n) + 6*(18270*n^7-160080*n^6+500851*n^5-666969*n^4+307749*n^3+70849*n^2-76222*n+8288)*n*a(n-1) + 6*(54810*n^8-562455*n^7+2191158*n^6-3956204*n^5+2960986*n^4+88959*n^3-1045774*n^2+187688*n+69888)*a(n-2) + 6*(109620*n^8-1289340*n^7+5897421*n^6-13016841*n^5+13725877*n^4-5967199*n^3+2484230*n^2-3359528*n+1002624)*a(n-3) - 6*(54810*n^8-726885*n^7+3719313*n^6-9080919*n^5+10367473*n^4-4378276*n^3+1152956*n^2-2297912*n+768096)*a(n-4) + (3*n-7)*(3*n-12)*(3*n-14)*(3045*n^5-11455*n^4+8631*n^3+1507*n^2+2376*n-1368)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 2 and a(4) = 0. (End)
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MAPLE
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add( (-1)^k*binomial(n+2*k+1, k) * binomial(n+2*k+1, n-k)/(n+2*k+1), k=0..n) ;
end proc:
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PROG
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(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+2*k+1, k)*binomial(n+2*k+1, n-k)/(n+2*k+1));
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CROSSREFS
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KEYWORD
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sign,easy,changed
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AUTHOR
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STATUS
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approved
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A364376
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G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^4).
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+10
2
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1, 0, -1, 3, -4, -9, 73, -212, 111, 1956, -10078, 21466, 29823, -418183, 1561911, -1722963, -13205004, 86962328, -232448945, -109578204, 3849218852, -17135183489, 27800381006, 113891855632, -966644138742, 3075070731677, -833503324311, -41673632701038
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OFFSET
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0,4
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FORMULA
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a(n) = Sum_{k=0..n} (-1)^k * binomial(n+3*k+1,k) * binomial(n+3*k+1,n-k) / (n+3*k+1).
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PROG
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(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+3*k+1, k)*binomial(n+3*k+1, n-k)/(n+3*k+1));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A366115
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Expansion of (1/x) * Series_Reversion( x*(1+x+x^2)/(1+x)^5 ).
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+10
2
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1, 4, 21, 125, 801, 5390, 37558, 268656, 1961355, 14555266, 109472688, 832625469, 6393072182, 49488174700, 385795571040, 3026190911853, 23867383581009, 189156323865632, 1505649098866535, 12031665674394905, 96486323017581420, 776255276240140980
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OFFSET
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0,2
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FORMULA
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a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(4*n-k+4,n-2*k).
P-recursive: 6*n*(2*n+3)*(n^2-1)*(2021*n^3-8037*n^2+9946*n-3672)*a(n) = 4*n*(n-1)*(94987*n^5-282752*n^4+140624*n^3+146936*n^2-56373*n-18522)*a(n-1) - 6*(n-1)*(367822*n^6-1646645*n^5+2610582*n^4-1674935*n^3+259948*n^2+125940*n-35712)*a(n-2) + 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(2021*n^3-1974*n^2-65*n+258)*a(n-3) with a(0) = 1, a(1) = 4 and a(2) = 21.
G.f. A(x) satisfies 1 + x*A(x) = (1/x) * series_reversion( x/c(x*c(x)) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
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MAPLE
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seq(simplify(1/(n+1)*binomial(4*n+4, n)*hypergeom([n+1, -(1/2)*n, (1/2)*(1-n)], [3*n+5, -4*(n+1)], 4)), n = 0..20); # Peter Bala, Aug 22 2024
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PROG
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(PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(n+k, k)*binomial(4*n-k+4, n-2*k))/(n+1);
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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A366114
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Expansion of (1/x) * Series_Reversion( x*(1+x+x^2)/(1+x)^3 ).
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+10
1
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1, 2, 4, 7, 9, 2, -34, -130, -284, -284, 730, 4864, 14860, 27134, 6462, -170865, -771303, -2005828, -2751028, 3491747, 36288137, 130265102, 283131062, 210905402, -1317613954, -7461822262, -22297519418, -38398674146, 10151248222, 355843715494, 1495838414326
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(2*n-k+2,n-2*k).
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PROG
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(PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(n+k, k)*binomial(2*n-k+2, n-2*k))/(n+1);
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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