A188682
Partial sums of binomials bin(3n,n)^2/(2n+1).
Original entry on oeis.org
1, 4, 49, 1057, 28282, 848101, 27357493, 928760053, 32747441926, 1188869998801, 44174723634526, 1672716549215326, 64340599136306926, 2507814491482180894, 98859670298036582494, 3935425516392739090270, 158006444406545953115743
Offset: 0
Cf.
A005809,
A001764,
A188676,
A104859,
A188678,
A188679,
A188680,
A188681,
A188683,
A188684,
A188685,
A188686,
A188687.
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Table[Sum[Binomial[3k,k]^2/(2k+1),{k,0,n}],{n,0,20}]
Accumulate[Table[Binomial[3n,n]^2/(2n+1),{n,0,20}]] (* Harvey P. Dale, Jul 10 2016 *)
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makelist(sum(binomial(3*k,k)^2/(2*k+1),k,0,n),n,0,20);
A345368
a(n) = Sum_{k=0..n} binomial(5*k,k) / (4*k + 1).
Original entry on oeis.org
1, 2, 7, 42, 327, 2857, 26608, 258488, 2588933, 26539288, 277082658, 2936050788, 31494394563, 341325970323, 3731742758203, 41108999917483, 455850863463768, 5084213586320193, 56997201842602368, 641906808539396253, 7258985455500009623, 82393287049581399283
Offset: 0
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Table[Sum[Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^4 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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a(n) = sum(k=0, n, binomial(5*k, k)/(4*k+1)); \\ Michel Marcus, Jul 28 2021
A346065
a(n) = Sum_{k=0..n} binomial(6*k,k) / (5*k + 1).
Original entry on oeis.org
1, 2, 8, 59, 565, 6046, 68878, 818276, 10021910, 125629220, 1603943486, 20783993414, 272641113110, 3613484662965, 48313969712685, 650888627139801, 8826840286257595, 120398870546499685, 1650711840886884265, 22735860619151166130, 314441081323870331656
Offset: 0
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Table[Sum[Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^5 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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a(n) = sum(k=0, n, binomial(6*k, k)/(5*k+1)); \\ Michel Marcus, Jul 28 2021
A346671
a(n) = Sum_{k=0..n} binomial(7*k,k) / (6*k + 1).
Original entry on oeis.org
1, 2, 9, 79, 898, 11370, 153148, 2150836, 31140511, 461462144, 6964815000, 106691488130, 1654539334220, 25923944408960, 409770113121064, 6526344613981944, 104632592920840659, 1687270854882480906, 27348675382672733281, 445328790513987869681, 7281393330439106226281
Offset: 0
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Table[Sum[Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^6 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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a(n) = sum(k=0, n, binomial(7*k, k)/(6*k+1)); \\ Michel Marcus, Jul 28 2021
A346672
a(n) = Sum_{k=0..n} binomial(8*k,k) / (7*k + 1).
Original entry on oeis.org
1, 2, 10, 102, 1342, 19620, 305004, 4943352, 82595376, 1412486081, 24602515801, 434935956337, 7783978950825, 140752989839105, 2567623696254905, 47195200645619009, 873239636055018809, 16251426606785706209, 304007720310330530081, 5713101394865420846381
Offset: 0
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Table[Sum[Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^7 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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a(n) = sum(k=0, n, binomial(8*k, k)/(7*k+1)); \\ Michel Marcus, Jul 28 2021
A345367
a(n) = Sum_{k=0..n} binomial(4*k,k) / (3*k + 1).
Original entry on oeis.org
1, 2, 6, 28, 168, 1137, 8221, 62041, 482773, 3845033, 31188921, 256757719, 2139691083, 18015030073, 153008796673, 1309402039993, 11279339531413, 97724562251137, 851035285261745, 7445189624293545, 65401191955640665, 576639234410182210, 5101317352349364430
Offset: 0
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Table[Sum[Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^3 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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a(n) = sum(k=0, n, binomial(4*k, k)/(3*k+1)); \\ Michel Marcus, Jul 28 2021
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