A340986 -id:A340986 - cp's OEIS frontend

A000275 Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden.

1, 1, 3, 19, 211, 3651, 90921, 3081513, 136407699, 7642177651, 528579161353, 44237263696473, 4405990782649369, 515018848029036937, 69818743428262376523, 10865441556038181291819, 1923889742567310611949459, 384565973956329859109177427, 86180438505835750284241676121

Offset: 0

N. J. A. Sloane

a(n) has the Lucas property, namely a(n) is congruent to a(n_0)a(n_1)...a(n_k) modulo p for any prime p where n_0,n_1,... are the base p digits of n. (Carlitz via McIntosh)

			From _Peter Bala_, Aug 08 2011: (Start)
a(3) = 19: The 19 pairs of permutations in the group S_3 x S_3 with no common rises correspond to the zero entries in the table below.
  ======================================
   Number of common rises in S_3 x S_3
  ======================================
     | 123   132   213   231   312   321
  ======================================
  123|  2     1     1     1     1     0
  132|  1     1     0     1     0     0
  213|  1     0     1     0     1     0
  231|  1     1     0     1     0     0
  312|  1     0     1     0     1     0
  321|  0     0     0     0     0     0
(End)
G.f. = 1 + x + 3*x^2 + 19*x^3 + 211*x^4 + 3651*x^5 + 90921*x^6 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..261
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (8) (with t=0 and m=2) on p. 249.
Leonid Bedratyuk and Nataliia Luno, Connection problems for the generalized hypergeometric Appell polynomials, Carpathian Math. Publ. (2020) Vol. 12, No. 1, 10-18.
L. Carlitz, The coefficients of the reciprocal of J_0(x), Archiv. Math. 6 (1955), 121-127.
L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884.
L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884. [Annotated scanned copy]
L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975.
Jan Geuenich, Tilting modules for the Auslander algebra of K[x]/(xn), arXiv:1803.10707 [math.RT], 2018.
Gunnar Thor Magnússon, The inner product on exterior powers of a complex vector space, arXiv preprint arXiv:1401.4048 [math.AG], 2014.
R. McIntosh, A generalization of a congruential property of Lucas, Amer. Math. Monthly 99(3) (1992), 231-238; see page 232. MR1216210 (95b:11008)
J. Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498.
Jonathan D. H. Smith, Commutative Moufang loops and Bessel functions, Invent. Math. 67(1) (1982), 173-187.
Index entries for sequences related to Bessel functions or polynomials

Row 2 of A212855.
Cf. A055133 (absolute value of column 0 of triangle), A192721 (column 1), A115368.
Column k=1 of A340986.

A000275 := proc(n) sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): n!^2*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:
seq(A000275(n), n=0..17); # Peter Luschny, May 27 2017

a[0] = 1; a[n_] := a[n] = Sum[(-1)^(r+n+1)*Binomial[n, r]^2 a[r], {r, 0, n-1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 05 2013 *)
CoefficientList[Series[1/BesselJ[0,Sqrt[4*x]], {x, 0, 20}], x]* Range[0, 20]!^2 (* Vaclav Kotesovec, Mar 02 2014 *)
a[ n_] := If[ n < 0, 0, (n! 2^n)^2 SeriesCoefficient[ 1 / BesselJ[ 0, x], {x, 0, 2 n}]]; (* Michael Somos, Aug 20 2015 *)

{a(n) = if( n<0, 0, n!^2 * 4^n * polcoeff( 1 / besselj(0, x + x * O(x^(2*n))), 2*n))}; /* Michael Somos, May 17 2004 */

a(n) = Sum_{r=0..n-1} (-1)^(r+n+1) binomial(n, r)^2 a(r), if n > 0.
Sum_{n>=0} a(n) * x^n / n!^2 = 1 / J_0(sqrt(4*x)). - _Michael Somos, May 17 2004
From Peter Bala, Aug 08 2011: (Start)
Conjectural formula: 1 = Sum_{n>=0} a(n)*x^n*Sum_{k>=0} binomial(n+k,k)^2*(-x)^k.
Apart from the initial term, first column of A192721. (End)
E.g.f.: 1/J_0(sqrt(4*x)) = 1 + x/Q(0), where Q(k) = (k+1)^2 - x + (k+1)^2*x/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2013
a(n) ~ c * (n!)^2 / r^n, where r = (1/4)*BesselJZero[0,1]^2 = 1.4457964907366961302939989396139517587678604516... and c = 1/(sqrt(r) * BesselJ(1, 2*sqrt(r))) = 1.60197469692804662664846689139151227422675123376219... - Vaclav Kotesovec, Mar 02 2014, updated Apr 01 2018

More terms from Christian G. Bower, Apr 25 2000

A336271 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * binomial(2k,k) a(n-k).

Original entry on oeis.org

1, 2, 10, 92, 1354, 29252, 873964, 34555880, 1748176714, 110183215988, 8467704986260, 779536758060920, 84699429189141100, 10725613123706081720, 1565870044943751242440, 261092436660169105108592, 49312362996510562406915914, 10473104312824253527997052500

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Cf. A000275, A000984, A002893.

Column k=2 of A340986.

a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 Binomial[2 k, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^2.

a(n) ~ (n!)^2 * n / (BesselJ(1, 2*sqrt(r))^2 * r^(n+1)), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 15 2020

A336638 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^3.

Original entry on oeis.org

1, 3, 21, 255, 4725, 123903, 4368729, 199467243, 11455187445, 808475761695, 68805857523321, 6950458374996843, 822292004658568761, 112639503374757412875, 17688916392275574761805, 3157133540377493872350855, 635546443798928578953138165

Offset: 0

Ilya Gutkovskiy, Jul 28 2020

nonn

Cf. A000275, A002893, A336271, A336639.

Column k=3 of A340986.

nmax = 16; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^3, {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 HypergeometricPFQ[{1/2, -k, -k}, {1, 1}, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * A002893(k) * a(n-k).

a(n) ~ n!^2 * n^2 / (2 * r^(n + 3/2) * BesselJ(1, 2*sqrt(r))^3), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 11 2025

A336639 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^4.

Original entry on oeis.org

1, 4, 36, 544, 12196, 377904, 15438816, 803602944, 51908768676, 4074743122384, 382079412133936, 42184889139337344, 5417567866536188896, 800808722921088352384, 135006904500993157933056, 25751088570167886107910144, 5517695042810314282550432676

Offset: 0

Ilya Gutkovskiy, Jul 28 2020

nonn

In general, if k>=1 and Sum_{n>=0} a(n) * x^n / n!^2 = 1 / BesselJ(0, 2*sqrt(x))^k, then a(n) ~ n!^2 * n^(k-1) / ((k-1)! * r^(n + k/2) * BesselJ(1, 2*sqrt(r))^k), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4. - Vaclav Kotesovec, Jul 11 2025

Cf. A000275, A002895, A336271, A336638.

Column k=4 of A340986.

nmax = 16; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^4, {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 Binomial[2 k, k] HypergeometricPFQ[{1/2, -k, -k, -k}, {1, 1, 1/2 - k}, 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * A002895(k) * a(n-k).

a(n) ~ n!^2 * n^3 / (6 * r^(n+2) * BesselJ(1, 2*sqrt(r))^4), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 11 2025

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

Ilya Gutkovskiy, Jul 29 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..99

Cf. A000275, A033935, A336271, A336638, A336639.

Main diagonal of A340986.

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}]

a(n) ~ c * d^n * n!^2 / sqrt(n), where d = 3.431031961004073074179854315227049823720211... and c = 0.31156343453490677011135864540173577785263... - Vaclav Kotesovec, May 04 2024

cp's OEIS Frontend

A000275 Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden.

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A336271 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * binomial(2*k,k) * a(n-k).

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A336638 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^3.

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A336639 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^4.

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A336665 a(n) = (n!)^2 * [x^n] 1 / BesselJ(0,2*sqrt(x))^n.

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Formula

A336271 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * binomial(2k,k) a(n-k).