A336636
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^3 - 1).
Original entry on oeis.org
1, 3, 33, 660, 20817, 935388, 56149098, 4311694467, 410200118577, 47174279349540, 6431874002292978, 1023398757621960327, 187566773426941146498, 39164789611542644630415, 9229712819952662426436507, 2435069724188535096598261305
Offset: 0
-
nmax = 15; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]]^3 - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 HypergeometricPFQ[{1/2, -k, -k}, {1, 1}, 4] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
A367177
Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).
Original entry on oeis.org
1, 1, 2, 1, 8, 6, 1, 18, 54, 20, 1, 32, 216, 320, 70, 1, 50, 600, 2000, 1750, 252, 1, 72, 1350, 8000, 15750, 9072, 924, 1, 98, 2646, 24500, 85750, 111132, 45276, 3432, 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 1, 2;
[2] 1, 8, 6;
[3] 1, 18, 54, 20;
[4] 1, 32, 216, 320, 70;
[5] 1, 50, 600, 2000, 1750, 252;
[6] 1, 72, 1350, 8000, 15750, 9072, 924;
[7] 1, 98, 2646, 24500, 85750, 111132, 45276, 3432;
[8] 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870;
[9] 1, 162, 7776, 141120, 1111320, 4000752, 6519744, 4447872, 1042470, 48620;
-
p := n -> hypergeom([1/2, -n, -n], [1, 1], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
A318397
Triangle read by rows: T(n,k) = binomial(n,k)^2 * binomial(2*(n-k), n-k).
Original entry on oeis.org
1, 2, 1, 6, 8, 1, 20, 54, 18, 1, 70, 320, 216, 32, 1, 252, 1750, 2000, 600, 50, 1, 924, 9072, 15750, 8000, 1350, 72, 1, 3432, 45276, 111132, 85750, 24500, 2646, 98, 1, 12870, 219648, 724416, 790272, 343000, 62720, 4704, 128, 1, 48620, 1042470, 4447872, 6519744, 4000752, 1111320, 141120, 7776, 162, 1
Offset: 0
Triangle begins:
1
2 1
6 8 1
20 54 18 1
70 320 216 32 1
...
-
T[ n_, k_] := Binomial[n, k]^2 Binomial[2 n - 2 k, n - k];
-
{T(n, k) = binomial(n, k)^2 * binomial(2*(n-k), n-k)};
A357770
Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-3 node.
Original entry on oeis.org
1, 3, 30, 372, 5112, 74448, 1125408, 17461440, 276193152, 4433878272, 72022049280, 1181146106880, 19524892723200, 324921616773120, 5438136568504320, 91467357685235712, 1545090682931085312, 26199310348842762240, 445746455962332561408, 7606624602795641929728
Offset: 0
a(2)=30, because there are 3*3=9 paths that visit one of three adjacent vertices, return to the origin, and again visit an adjacent vertex and return to the origin; 3*5=15 paths visiting one of five distance-2 vertices that are adjacent to the three adjacent vertices; plus 3*2=6 paths traversing the perimeter of three adjacent rhombi in counterclockwise or clockwise direction; all resulting in a closed path of length 2n=2*2=4.
The accompanying sequences for the number of paths that return to a degree-6 node is
A357771.
-
a[0] := 1; a[n_] := Sum[Binomial[n, k]*Sum[Binomial[n, j]*Sum[(1/(2^(j+1)))*Binomial[2*i, j]*Binomial[2*i, i]*Binomial[2*(j-i), j-i], {i, 0, j}], {j, 0, n}], {k, 0, n}]; Flatten[Table[a[n], {n, 0, 19}]] (* Detlef Meya, May 20 2024 *)
A357771
Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-6 node.
Original entry on oeis.org
1, 6, 60, 744, 10224, 148896, 2250816, 34922880, 552386304, 8867756544, 144044098560, 2362292213760, 39049785446400, 649843233546240, 10876273137008640, 182934715370471424, 3090181365862170624, 52398620697685524480, 891492911924665122816, 15213249205591283859456, 260315328935885892747264
Offset: 0
a(2)=60, because there are 6*6=36 paths that visit one of six adjacent vertices, return to the origin, and again visit an adjacent vertex and return to the origin; plus 6*4=24 paths that pass through one of the six vertices at distance 2, leaving and returning via any of two available paths to that vertex; all resulting in a closed path of length 2n=2*2=4.
The accompanying sequences for the number of paths that return to a degree-3 node is
A357770.
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a[n_] := Sum[Binomial[n, k]*Sum[Binomial[n, j]*Sum[(1/(2^j))*Binomial[2*i, j]*Binomial[2*i, i]*Binomial[2*(j-i), j-i],{i,0,j}],{j,0,n}],{k,0,n}]; Flatten[Table[a[n],{n,0,17}]] (* Detlef Meya, May 15 2024 *)
-
a(n) = sum(k=0, n, binomial(n, k) * sum(j=0, n, binomial(n, j) * sum(i=0, j, (1/(2^j)*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i))))); \\ Michel Marcus, May 20 2024
A357810
Number of n-step closed paths on the Cairo pentagonal lattice graph starting from a degree-4 node.
Original entry on oeis.org
1, 0, 4, 0, 24, 8, 164, 136, 1236, 1704, 10116, 19144, 88616, 205208, 818764, 2155160, 7873440, 22463400, 77954740, 233894600, 788314984, 2440865400, 8095906076, 25569342520, 84107990356, 269034666280
Offset: 0
For n=2, the a(2) = 4 solutions visit one of the four vertices adjacent to the initial vertex, and then return.
For n=5, the a(5) = 8 solutions consist of counterclockwise or clockwise traversals of the circumference of any of the four pentagons that surround the initial vertex.
The accompanying sequences for the number of paths that return to a degree-3 node is
A357811.
A357811
Number of n-step closed paths on the Cairo pentagonal lattice graph starting from a degree-3 node.
Original entry on oeis.org
1, 0, 3, 0, 17, 6, 115, 100, 867, 1236, 7117, 13770, 62545, 146866, 579387, 1537920, 5581725, 16002810, 55329435, 166465820, 559913787, 1736268432, 5752600961, 18182999274, 59777071435, 191287075320
Offset: 0
For n=2, the a(2) = 3 solutions visit one of the three vertices adjacent to the initial vertex, and then return.
For n=5, the a(5) = 6 solutions consist of counterclockwise or clockwise traversals of the circumference of any of the three pentagons that surround the initial vertex.
The accompanying sequences for the number of paths that return to a degree-4 node is
A357810.
A291277
Primes p such that p does not divide any term of the Apery-like sequence A081085.
Original entry on oeis.org
3, 11, 17, 19, 43, 59, 73, 83, 89, 107, 179, 211, 227, 233, 241, 257, 307, 331, 337, 379, 401, 409, 419, 433, 449, 457, 467, 521, 547, 563, 577, 587, 593, 601, 619, 641, 643, 683, 691, 739, 761, 769, 811, 827, 859, 881, 883, 929, 937, 947, 953
Offset: 1
For primes that do not divide the terms of the sequences
A000172,
A005258,
A002893,
A081085,
A006077,
A093388,
A125143,
A229111,
A002895,
A290575,
A290576,
A005259 see
A260793,
A291275-
A291284 and
A133370 respectively.
A291278
Primes p such that p does not divide any term of the Apery-like sequence A006077.
Original entry on oeis.org
2, 5, 13, 17, 29, 37, 41, 61, 73, 97, 101, 113, 137, 149, 157, 173, 181, 197, 229, 241, 257, 277, 313, 317, 349, 353, 389, 409, 421, 433, 449, 457, 461, 509, 541, 569, 577, 593, 613, 641, 653, 661, 673, 709, 757, 761, 769, 797, 809, 829, 853, 857
Offset: 1
For primes that do not divide the terms of the sequences
A000172,
A005258,
A002893,
A081085,
A006077,
A093388,
A125143,
A229111,
A002895,
A290575,
A290576,
A005259 see
A260793,
A291275-
A291284 and
A133370 respectively.
A291279
Primes p such that p does not divide any term of the Apery-like sequence A093388.
Original entry on oeis.org
5, 11, 29, 31, 59, 79, 107, 131, 149, 151, 173, 179, 193, 197, 199, 241, 251, 271, 317, 409, 433, 439, 443, 457, 461, 509, 557, 587, 601, 607, 659, 677, 701, 727, 751, 769, 773, 797, 821, 823, 827, 919, 971, 1009, 1013, 1019, 1033, 1039, 1061, 1063, 1087
Offset: 1
For primes that do not divide the terms of the sequences
A000172,
A005258,
A002893,
A081085,
A006077,
A093388,
A125143,
A229111,
A002895,
A290575,
A290576,
A005259 see
A260793,
A291275-
A291284 and
A133370 respectively.
Comments