A336665
a(n) = (n!)^2 * [x^n] 1 / BesselJ(0,2*sqrt(x))^n.
Original entry on oeis.org
1, 1, 10, 255, 12196, 939155, 106161756, 16554165495, 3404986723720, 893137635015219, 290965846152033460, 115256679181251696803, 54552992572663333862400, 30406695393635479756804525, 19712738332895648545008815416, 14707436666152282009334357074335
Offset: 0
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Table[(n!)^2 SeriesCoefficient[1/BesselJ[0, 2 Sqrt[x]]^n, {x, 0, n}], {n, 0, 15}]
A287316[n_, k_] := A287316[n, k] = If[n == 0, 1, If[k < 1, 0, Sum[Binomial[n, j]^2 A287316[n - j, k - 1], {j, 0, n}]]]; b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[(-1)^(j + 1) Binomial[n, j]^2 A287316[j, k] b[n - j, k], {j, 1, n}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 15}]
A287315
Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287316.
Original entry on oeis.org
1, 0, 1, 0, 1, 3, 0, 1, 16, 19, 0, 1, 65, 299, 211, 0, 1, 246, 3156, 7346, 3651, 0, 1, 917, 28722, 160322, 237517, 90921, 0, 1, 3424, 245407, 2864912, 9302567, 9903776, 3081513, 0, 1, 12861, 2041965, 46261609, 288196659, 632274183, 520507423, 136407699
Offset: 0
Triangle starts:
0: [1]
1: [0, 1]
2: [0, 1, 3]
3: [0, 1, 16, 19]
4: [0, 1, 65, 299, 211]
5: [0, 1, 246, 3156, 7346, 3651]
6: [0, 1, 917, 28722, 160322, 237517, 90921]
7: [0, 1, 3424, 245407, 2864912, 9302567, 9903776, 3081513]
...
Let q4(x) = (x + 65*x^2 + 299*x^3 + 211*x^4) / (1-x)^5 then the coefficients of the series expansion of q4 give A169712, which is column 4 of A287316.
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Delta := proc(a, n) local del, A, u;
A := [seq(a(j), j=0..n+1)]; del := (a, k) -> `if`(k=0, a(0), a(k)-a(k-1));
for u from 0 to n do A := [seq(del(k -> A[k+1], j), j=0..n)] od end:
A287315_row := n -> Delta(A287314_poly(n), n):
for n from 0 to 7 do A287315_row(n) od;
A287315_eulerian := (n,x) -> add(A287315_row(n)[k+1]*x^k,k=0..n)/(1-x)^(n+1):
for n from 0 to 4 do A287315_eulerian(n,x) od;
A334412
Number of ordered pairs of permutations of [n] avoiding synchronous double descent pairs.
Original entry on oeis.org
1, 1, 4, 35, 545, 13250, 463899, 22106253, 1375915620, 108386009099, 10540705282001, 1240370638524842, 173704235075714947, 28549174106487593365, 5441843626292088857818, 1190762128123996264128849, 296456799935194225886732961, 83321234634397591315509479058
Offset: 0
a(3) = (3!)^2 - 1 = 35: only (321,321) does not avoid synchronous double descent pairs among the ordered pairs of permutations of [3].
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b:= proc(n, u, v, t) option remember; `if`(n=0, 1, add(add(
b(n-1, sort([u-j, v-i])[], 1), i=1..v)+add(
b(n-1, sort([u-j, v+i-1])[], 1), i=1..n-v), j=1..u)+add(add(
b(n-1, sort([u+j-1, v-i])[], 1), i=1..v)+add(`if`(t=0, 0,
b(n-1, sort([u+j-1, v+i-1])[], 0)), i=1..n-v), j=1..n-u))
end:
a:= n-> b(n$3, 1):
seq(a(n), n=0..21);
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nn = 20; a=Apply[Plus,Table[Normal[Series[y x^3/(1 - y x - y x^2), {x, 0, nn}]][[n]]/(n +2)!^2, {n, 1, nn - 2}]] /. y -> -1;Map[Select[#, # > 0 &] &,
Range[0, nn]!^2 CoefficientList[Series[1/(1 - x - a), {x, 0, nn}], {x, y}]] // Flatten (* Geoffrey Critzer, Apr 27 2020 *)
A336606
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) / BesselJ(0,2*sqrt(x)).
Original entry on oeis.org
1, 2, 9, 70, 851, 15246, 384147, 13065354, 578905875, 32440563766, 2243907466283, 187796863841346, 18704441632101337, 2186374265471576090, 296396762529435076953, 46126320892158605384334, 8167358455139620845210003, 1632571811017090501346518086
Offset: 0
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nmax = 17; CoefficientList[Series[Exp[x]/BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
A000275[0] = 1; A000275[n_] := A000275[n] = -Sum[(-1)^(n - k) Binomial[n, k]^2 A000275[k], {k, 0, n - 1}]; a[n_] := n! Sum[Binomial[n, k] A000275[k]/k!, {k, 0, n}]; Table[a[n], {n, 0, 17}]
A336608
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(-x) / BesselJ(0,2*sqrt(x)).
Original entry on oeis.org
1, 0, 1, 4, 51, 856, 21435, 725796, 32132499, 1800176176, 124511280723, 10420458131260, 1037868062069113, 121317006426807192, 16446390218708245393, 2559445829942874207804, 453188354421968867989395, 90587738500599611033753184
Offset: 0
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nmax = 17; CoefficientList[Series[Exp[-x]/BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
A000275[0] = 1; A000275[n_] := A000275[n] = -Sum[(-1)^(n - k) Binomial[n, k]^2 A000275[k], {k, 0, n - 1}]; a[n_] := n! Sum[(-1)^(n - k) Binomial[n, k] A000275[k]/k!, {k, 0, n}]; Table[a[n], {n, 0, 17}]
A298595
G.f.: Sum_{n>=0} a(n)*x^(2*n)/((2*n)!)^2 = 1/BesselJ(0,x).
Original entry on oeis.org
1, 1, 27, 4275, 2326275, 3260434275, 9824561849025, 56272951734424425, 560476093710119461875, 9074718916938795106861875, 226586114542199918676706160625, 8362768986063791790897266120885625, 440616849129306857329147873116900455625, 32189976281042425371050387695609814928515625
Offset: 0
1/BesselJ(0,x) = 1 + x^2/(2!)^2 + 27*x^4/(4!)^2 + 4275*x^6/(6!)^2 + 2326275*x^8/(8!)^2 + ...
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nmax = 13; Table[(CoefficientList[Series[1/BesselJ[0, x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!^2)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 13; Table[(CoefficientList[Series[1/Hypergeometric0F1[1, -x^2/4], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!^2)[[n]], {n, 1, 2 nmax + 1, 2}]
A336609
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(1 / BesselJ(0,2*sqrt(x)) - 1).
Original entry on oeis.org
1, 1, 5, 52, 917, 24396, 909002, 45062697, 2862532213, 226403027044, 21794813189810, 2507115921526437, 339421509956163362, 53393907140415300317, 9653668439939308357991, 1987242385193691443059527, 461955240782446199029195253
Offset: 0
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nmax = 16; CoefficientList[Series[Exp[1/BesselJ[0, 2 Sqrt[x]] - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
A000275[0] = 1; A000275[n_] := A000275[n] = -Sum[(-1)^(n - k) Binomial[n, k]^2 A000275[k], {k, 0, n - 1}]; a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k A000275[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]