A000275 -id:A000275 - cp's OEIS frontend

A212860 Number of 7 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.

1, 1, 127, 275563, 4479288703, 347190069843751, 96426023622482278621, 78785944892341703819175577, 163925632052722656731213188429183, 777880066963402408939826643081996101263, 7717574897043522397037273525233635595811018377

Offset: 0

R. H. Hardin, May 28 2012

nonn

From Petros Hadjicostas, Sep 08 2019: (Start)
We generalize Daniel Suteu's recurrence from A212856. Notice first that, in the notation of Abramson and Promislow (1978), we have a(n) = R(m=7, n, t=0).
Letting y=0 in Eq. (8), p. 249, of Abramson and Promislow (1978), we get 1 + Sum_{n >= 1} R(m,n,t=0)*x^n/(n!)^m = 1/f(-x), where f(x) = Sum_{i >= 0} (x^i/(i!)^m). Matching coefficients, we get Sum_{s = 1..n} R(m, s, t=0) * (-1)^(s-1) * binomial(n,s)^m = 1, from which the recurrence in the Formula section follows.
(End)

			Some solutions for n=3:
  0 1 2   0 1 2   0 2 1   0 1 2   0 2 1   0 2 1   0 2 1
  1 2 0   0 2 1   0 2 1   1 0 2   0 2 1   1 0 2   2 1 0
  1 0 2   2 1 0   2 0 1   0 1 2   2 0 1   1 0 2   1 2 0
  0 2 1   1 0 2   0 2 1   1 0 2   0 1 2   2 0 1   0 1 2
  2 0 1   2 1 0   1 0 2   2 1 0   1 2 0   0 1 2   1 2 0
  2 1 0   0 1 2   1 0 2   0 1 2   2 0 1   1 0 2   2 1 0
  1 2 0   2 1 0   0 1 2   0 2 1   2 1 0   2 0 1   2 0 1

Seiichi Manyama, Table of n, a(n) for n = 0..92 (terms n=1..19 from R. H. Hardin)
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (8) on p. 249.

Row 7 of A212855.

Cf. A000012, A000225, A000275, A212850, A212851, A212852, A212853, A212854, A212856, A212857, A212858, A212859.

A212860 := proc(n) sum(z^k/k!^7, k = 0..infinity);
series(%^x, z=0, n+1): n!^7*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:
seq(A212860(n), n=1..10); # Peter Luschny, May 27 2017

T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k - j], {j, 1, k}]];
a[n_] := T[7, n];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Apr 01 2024, after Alois P. Heinz in A212855 *)

a(n) = (-1)^(n-1) + Sum_{s = 1..n-1} a(s) * (-1)^(n-s-1) * binomial(n,s)^m for n >= 2 with a(1) = 1. Here m = 7. - Petros Hadjicostas, Sep 08 2019

a(n) = (n!)^7 * [x^n] 1 / (1 + Sum_{k>=1} (-x)^k / (k!)^7). (see Petros Hadjicostas's comment on Sep 08 2019) - Seiichi Manyama, Jul 18 2020

a(0)=1 prepended by Seiichi Manyama, Jul 18 2020

A055133 Matrix inverse of A008459 (squares of entries of Pascal's triangle).

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -19, 27, -9, 1, 211, -304, 108, -16, 1, -3651, 5275, -1900, 300, -25, 1, 90921, -131436, 47475, -7600, 675, -36, 1, -3081513, 4455129, -1610091, 258475, -23275, 1323, -49, 1, 136407699, -197216832, 71282064, -11449536, 1033900, -59584, 2352, -64, 1

Offset: 0

Christian G. Bower, Apr 25 2000

Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0,2*sqrt(-y)). Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - Peter Bala, Jul 24 2013

			Table T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
      1;
     -1,       1;
      3,      -4,     1;
    -19,      27,    -9,     1;
    211,    -304,   108,   -16,   1;
  -3651,    5275, -1900,   300, -25,   1;
  90921, -131436, 47475, -7600, 675, -36, 1;
  ... [edited by _Petros Hadjicostas_, Aug 24 2019]
From _Peter Bala_, Jul 24 2013: (Start)
Function   |        Real zeros to 5 decimal places
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
R(5,x)     | 1, 5.40649,  7.23983
R(10,x)    | 1, 5.26894, 12.97405, 18.53109
R(15,x)    | 1, 5.26894, 12.94909, 24.04769, 33.87883
R(20,x)    | 1, 5.26894, 12.94909, 24.04216, 38.54959, 53.32419
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
E(alpha*x) | 1, 5.26894, 12.94909, 24.04216, 38.54835, 56.46772, ...
where alpha = -1.44579 64907 ... ( = -(A115365/2)^2).
Note: The n-th zero of E(alpha*x) may be calculated in Maple 17 using the instruction evalf( (BesselJZeros(0,n)/BesselJZeros(0,1))^2 ). (End)

Alois P. Heinz, Rows n = 0..99, flattened
J. Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498; see p. 493 with beta_{n,k} = |T(n,k)|.
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, 308(24) (2008), 6466-6500.

Cf. A000275, A008459 (matrix inverse), A115365.

T:= proc(n) local M;
       M:= Matrix(n+1, (i, j)-> binomial(i-1, j-1)^2)^(-1);
       seq(M[n+1, i], i=1..n+1)
    end:
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 14 2013

T[n_] := Module[{M}, M = Table[Binomial[i-1, j-1]^2, {i, 1, n+1}, {j, 1, n+1}] // Inverse; Table[M[[n+1, i]], {i, 1, n+1}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Nov 28 2015, after Alois P. Heinz *)

T(n, k) = (-1)^(n+k)*A000275(n-k)*C(n, k)^2.
From Peter Bala, Jul 24 2013: (Start)
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0,2*sqrt(-y)). Generating function: E(x*y)/E(y) = 1 + (-1 + x)*y + (3 - 4*x + x^2)*y^2/2!^2 + (-19 + 27*x - 9*x^2 + x^3)*y^3/3!^2 + ....
The n-th power of this array has a generating function E(x*y)/E(y)^n. In particular, the matrix inverse A008459 has a generating function E(y)*E(x*y).
Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} binomial(n,k)^2*R(k,x) with initial value R(0,x) = 1.
There appears to be a connection between the zeros of the Bessel function E(x) and the real zeros of the row polynomials R(n,x). Let alpha denote the root of E(x) = 0 that is smallest in absolute magnitude. Numerically, alpha = -1.44579 64907 ... ( = -(A115365/2)^2). It appears that the real zeros of R(n,x) approach zeros of E(alpha*x) as n increases. A numerical example is given below. Indeed, it may be the case that lim_{n -> inf} R(n,x)/R(n,0) = E(alpha*x) for arbitrary complex x. (End)

A238390 E.g.f.: x / BesselJ(1, 2*x) (even powers only).

Original entry on oeis.org

1, 1, 4, 35, 546, 13482, 485892, 24108513, 1576676530, 131451399794, 13609184032808, 1712978776719938, 257612765775847132, 45620136452519144700, 9396239458048330569840, 2227147531572856811691105, 601916577165056911293330930, 183994483721828524163677628370

Offset: 0

Vaclav Kotesovec, Mar 01 2014

nonn

Aerated, the e.g.f. is e^(a.t) =  1/AC(i*t) = 1/[I_1(2i*t)/(it)] = 1/Sum_{n>=0} (-1)^n t^(2n) / [n!(n+1)!] = a_0 + a_2 t^2/2! + a_4 t^4/4! + ... = 1 + t^2/2! + 4 t^4/4! + 35 t^6/6! + ..., where AC(t) is the e.g.f. for the aerated Catalan numbers c_n of A126120 and I_n(t) are the modified Bessel functions of the first kind (i = sqrt(-1)). The signed, aerated sequence b_n = (i)^n a_n has the e.g.f. e^(b.t) = 1/AC(t) and, therefore, (i*a. + c.)^n = Sum_{k=0..n} binomial(n,k) i^k a_k c_(n-k) vanishes except for n=0 for which it's unity. - Tom Copeland, Jan 23 2016
With q(n) = A126120(n+1) and q(0) = 0, d(2n) =  (-1)^n A238390(n) and zero for odd arguments, and r(2n+1) = (-1)^n A180874(n+1) and zero for even arguments, then r(n) = (q. + d.)^n = Sum_{k=0..n} binomial(n,k) q(k) d(n-k), relating these sequences (and A000108) through binomial convolutions. Then also, (r. + c. + d.)^n = r(n). See A180874 for proofs and for relations to A097610. For quick reference, q = (0, 1, 0, 2, 0, 5, 0, 14, ..), d = (1, 0, -1, 0, 4, 0, -35, 0, ..), and r = (0, 1, 0, -1, 0, 5, 0, -56, ..). - Tom Copeland, Jan 28 2016
Aerated and signed, this sequence contains the moments m(n) of the Appell polynomial sequence UMT(n,h1,h2) that is the umbral compositional inverse of the Appell sequence of Motzkin polynomials MT(n,h1,h2) of A097610 with exp[x UMT(.,h1,h2)] = e^(x*h1) / AC(x*y) where y = sqrt(h2) and AC is defined above. UMT(n,h1,h2) = (m.y + h1)^n with (m.)^(2n) = m(2n) = (-1)^n A238390(n) and zero otherwise. Consequently, the associated lower triangular matrices A007318(n,k)*m(n-k) and A007318(n,k)*A126120(n-k) form an inverse pair (cf. also A133314), and MT(n,UMT(.,h1,h2),h2) = h1^n = UMT(n,MT(.,h1,h2),h2). - Tom Copeland, Jan 30 2016

G. C. Greubel, Table of n, a(n) for n = 0..98

Cf. A103365, A180874, A115369, A000275, A002190.

Cf. A000108, A007318, A097610, A126120, A133314.

S:= series(x/BesselJ(1,2*x),x,102):
seq((2*j)!*coeff(S,x,2*j),j=0..50); # Robert Israel, Jan 31 2016

Table[(CoefficientList[Series[x/BesselJ[1, 2*x], {x, 0, 40}], x] * Range[0, 40]!)[[n]], {n, 1, 41, 2}]

a(n) ~ c * (n!)^2 / (sqrt(n) * r^n), where r = BesselJZero[1, 1]^2/16 = 0.91762316513274332857623611, and c = 1/(Sqrt[Pi]*BesselJ[2, BesselJZero[1, 1]]) = 1.4008104828035425937394082168... - Vaclav Kotesovec, Mar 01 2014, updated Apr 01 2018

A047796 a(n) = Sum_{k=0..n} Stirling1(n,k)^2.

Original entry on oeis.org

1, 1, 2, 14, 194, 4402, 147552, 6838764, 418389078, 32639603798, 3161107700156, 372023906062756, 52280302234036252, 8645773770675973804, 1661888635268695003484, 367390786215560629372920, 92552610850186107484661670, 26356304249588730696338349990

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi and Vaclav Kotesovec, Table of n, a(n) for n = 0..250 (terms 0..41 from Vincenzo Librandi)

Cf. A000275, A047797, A342111.

List([0..20], n-> Sum([0..n], k-> Stirling1(n,k)^2 )); # G. C. Greubel, Aug 07 2019

[(&+[StirlingFirst(n,k)^2: k in [0..n]]): n in [0..10]]; // G. C. Greubel, Aug 07 2019

seq(add(stirling1(n, k)^2, k = 0..n), n = 0..20); # G. C. Greubel, Aug 07 2019

Table[Sum[StirlingS1[n,k]^2,{k,0,n}],{n,0,20}] (* Emanuele Munarini, Jul 04 2011 *)

makelist(sum(stirling1(n,k)^2,k,0,n),n,0,24); /* Emanuele Munarini, Jul 04 2011 */

a(n) = sum(k=0, n, stirling(n, k, 1)^2); \\ Michel Marcus, Mar 26 2016

[sum(stirling_number1(n,k)^2 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019

A287314 Triangle read by rows, the coefficients of the polynomials generating the columns of A287316.

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 4, -9, 6, 0, -33, 82, -72, 24, 0, 456, -1225, 1250, -600, 120, 0, -9460, 27041, -30600, 17700, -5400, 720, 0, 274800, -826336, 1011017, -661500, 249900, -52920, 5040, 0, -10643745, 33391954, -43471624, 31149496, -13524000, 3622080, -564480, 40320

Offset: 0

Peter Luschny, May 27 2017

The zeta polynomials for the poset P_n of ordered pairs (S,T) where S,T are subsets of [n] with |S| = |T| ordered component-wise by inclusion. - Geoffrey Critzer, Jan 22 2021

			Triangle starts:
[0] 1
[1] 0,      1
[2] 0,     -1,       2
[3] 0,      4,      -9,       6
[4] 0,    -33,      82,     -72,      24
[5] 0,    456,   -1225,    1250,    -600,    120
[6] 0,  -9460,   27041,  -30600,   17700,  -5400,    720
[7] 0, 274800, -826336, 1011017, -661500, 249900, -52920, 5040
...
For example let p4(x) = -33*x + 82*x^2 - 72*x^3 + 24*x^4 then p4(n) = A169712(n).

Cf. A287316, A000384 (p2), A169711 (p3), A169712 (p4), A169713 (p5).

Cf. A000275(n), A212855.

A287314_row := proc(n) local k; sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): n!^2*coeff(%,z,n); seq(coeff(%,x,k), k=0..n) end:
for n from 0 to 8 do print(A287314_row(n)) od;
A287314_poly := proc(n) local k, x; sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): unapply(n!^2*coeff(%, z, n), x) end:
for n from 0 to 7 do A287314_poly(n) od;

nn = 10; e[x_] := Sum[x^n/n!^2, {n, 0, nn}];
f[list_] := CoefficientList[InterpolatingPolynomial[Table[{i, list[[i]]}, {i, 1, nn}], m], m];Drop[Map[f,Transpose[Table[Table[n!^2, {n, 0, nn}] CoefficientList[
Series[e[x]^k, {x, 0, nn}], x], {k, 1, nn}]]], -1] // Grid (* Geoffrey Critzer, Jan 22 2021 *)

Sum_{k=0..n} abs(T(n,k)) = A000275(n) = A212855_row(2).

A340986 Square array read by descending antidiagonals. T(n,k) is the number of ways to separate the columns of an ordered pair of n-permutations (that have been written as a 2 X n array, one atop the other) into k cells so that no cell has a column rise. For n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 19, 0, 1, 4, 21, 92, 211, 0, 1, 5, 36, 255, 1354, 3651, 0, 1, 6, 55, 544, 4725, 29252, 90921, 0, 1, 7, 78, 995, 12196, 123903, 873964, 3081513, 0, 1, 8, 105, 1644, 26215, 377904, 4368729, 34555880, 136407699, 0

Offset: 0

Geoffrey Critzer, Feb 01 2021

A column rise (cf. A000275) means a pair of adjacent columns within a cell where each entry in the first column is less than the adjacent entry in the second column. The order of the columns cannot change. The cells are allowed to be empty.

			Square array T(n,k) begins:
  1,    1,     1,      1,      1,      1, ...
  0,    1,     2,      3,      4,      5, ...
  0,    3,    10,     21,     36,     55, ...
  0,   19,    92,    255,    544,    995, ...
  0,  211,  1354,   4725,  12196,  26215, ...
  0, 3651, 29252, 123903, 377904, 939155, ...

R. P. Stanley, Enumerative Combinatorics, Vol. I, Second Edition, Section 3.13.

Alois P. Heinz, Antidiagonals n = 0..100, flattened

Columns k=0-4 give: A000007, A000275, A336271, A336638, A336639.
Rows n=0-2 give: A000012, A001477, A014105.
Main diagonal gives A336665.
Cf. A212855, A192721, A334394.

T:= (n, k)-> n!^2*coeff(series(1/BesselJ(0, 2*sqrt(x))^k, x, n+1), x, n):
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Feb 02 2021

nn = 6; B[n_] := n!^2; e[x_] := Sum[x^n/B[n], {n, 0, nn}];
Table[Table[B[n], {n, 0, nn}] PadRight[CoefficientList[Series[e[-x]^-k, {x, 0, nn}], x], nn + 1], {k, 0, nn}] // Grid

Let E(x) = Sum_{n>=0} x^n/n!^2. Then Sum_{n>=0} T(n,k)*x^n/n!^2 = 1/E(-x)^k.
T(n,k) = (n!)^2 * [x^n] 1/BesselJ(0,2*sqrt(x))^k. - Alois P. Heinz, Feb 02 2021
For fixed k>=1, T(n,k) ~ 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 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 11 2025

A192722 T(n,k) = Sum of multinomial(n; n_1,n_2,...,n_k)^2, where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.

Original entry on oeis.org

1, 1, 4, 1, 18, 36, 1, 68, 432, 576, 1, 250, 3900, 14400, 14400, 1, 922, 32400, 252000, 648000, 518400, 1, 3430, 262542, 3880800, 19404000, 38102400, 25401600, 1, 12868, 2119152, 56664384, 493920000, 1795046400, 2844979200, 1625702400

Offset: 1

Peter Bala, Jul 11 2011

Compare with triangle A019538, whose entries are given by
... Sum multinomial(n; n_1,n_2,...,n_k), where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.
For related tables see A061691 and A192721.
Let P be the poset of all ordered pairs (S,T) of subsets of [n] with |S|=|T|, ordered componentwise by inclusion.  T(n,k) is the number of length k chains in P from ({},{}) to ([n],[n]). - Geoffrey Critzer, Apr 15 2020

			The triangle begins
n/k|..1.....2.......3........4........5........6
================================================
.1.|..1
.2.|..1.....4
.3.|..1....18.....36
.4.|..1....68.....432......576
.5.|..1...250....3900....14400....14400
.6.|..1...922...32400...252000...648000...518400
...
T(4,2) = 68:
There are 3 compositions of 4 into 2 parts, namely, 4 = 2 + 2 = 1 + 3 = 3 + 1; hence
T(4,2) = (4!/(2!*2!))^2 + (4!/(1!*3!))^2 + (4!/(3!*1!))^2
= 36 + 16 + 16 = 68.
Matrix identity: A192721 * Pascal's triangle = row reverse of A192722:
/...1................\ /..1..............\
|...3.....1...........||..1....1..........|
|..19....16.....5.....||..1....2....1.....|
|.211...299....65....1||..1....3....3....1|
|.....................||..................|
=
/...1...................\
|...4......1.............|
|..36.....18......1......|
|.576....432.....68.....1|
|........................|

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A001044, A002190, A061691, A192721, A102221 (row sums), A000275 (alternating row sums).

J := unapply(BesselJ(0, 2*sqrt(-1)*sqrt(z)), z):
G := 1/(1-x*(J(z)-1)):
Gser := simplify(series(G, z = 0, 15)):
for n from 1 to 14 do
P[n] := n!^2*sort(coeff(Gser, z, n)) od:
for n from 1 to 14 do seq(coeff(P[n], x, k), k = 1..n) od;
# yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(
      `if`(n=0, 1, add(x*b(n-i)/i!^2, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)*n!^2):
seq(T(n), n=1..14);  # Alois P. Heinz, Sep 10 2019

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[x b[n-i]/i!^2, {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n] n!^2];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

Generating function: Let J(z) = Sum_{n>=0} z^n/n!^2. Then
1 + Sum_{n>=1} (Sum_{k = 1..n} T(n,k)*x^k)*z^n/n!^2 = 1/(1 - x*(J(z) - 1))
  = 1 + x*z + (x + 4*x^2)*z^2/2!^2 + (x + 18*x^2 + 36*x^3)*z^3/3!^2 + ....
Relations with other sequences:
The change of variable z -> z/x followed by x -> 1/(x - 1) transforms the above bivariate generating function 1/(1 - x*(J(z) - 1)) into (1 - x)/(-x + J(z*(x-1))), which is the generating function for A192721.
1/k!*T(n,k) = A061691(n,k).
T(n,n) = n!^2 = A001044(n).
Row sums = A102221.
For n>=1, Sum_{k = 1..n} (-1)^(n+k)*T(n,k)/k = A002190(n).

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

cp's OEIS Frontend

A212860 Number of 7 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.

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A055133 Matrix inverse of A008459 (squares of entries of Pascal's triangle).

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A238390 E.g.f.: x / BesselJ(1, 2*x) (even powers only).

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A047796 a(n) = Sum_{k=0..n} Stirling1(n,k)^2.

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A287314 Triangle read by rows, the coefficients of the polynomials generating the columns of A287316.

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A340986 Square array read by descending antidiagonals. T(n,k) is the number of ways to separate the columns of an ordered pair of n-permutations (that have been written as a 2 X n array, one atop the other) into k cells so that no cell has a column rise. For n >= 0, k >= 0.

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A192722 T(n,k) = Sum of multinomial(n; n_1,n_2,...,n_k)^2, where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.

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