A023998 -id:A023998 - cp's OEIS frontend

A342196 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^2 * a(k-1).

1, 1, 5, 23, 155, 1355, 14371, 183911, 2781283, 48726355, 976903875, 22183097191, 565060532965, 16016170519017, 501714014484813, 17265124180702953, 649178961366102597, 26544344366333824055, 1175291769917975444817, 56133021061270139242637, 2881893164859601701738005

Offset: 0

Ilya Gutkovskiy, Mar 04 2021

nonn

Cf. A023998, A040027, A101514, A102221, A342197, A342198, A342199.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]

A343254 Triangle read by rows: T(n,k) is the number of 2-balanced partitions of a set of n elements in which the first and the second subsets have cardinality k, for n >= 0, k = 0..floor(n/2).

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 15, 5, 3, 52, 15, 8, 203, 52, 25, 16, 877, 203, 89, 53, 4140, 877, 354, 197, 131, 21147, 4140, 1551, 810, 512, 115975, 21147, 7403, 3643, 2193, 1496, 678570, 115975, 38154, 17759, 10201, 6697, 4213597, 678570, 210803, 93130, 51146, 32345, 22482

Offset: 0

Francesca Aicardi, Jun 04 2021

A 2-balanced partition is a partition of a set which is the union of three subsets, with the property that the cardinality of the first two subsets are equal (possibly zero), and each block contains the same number (possibly zero) of elements from the first and from the second subset. The rows add to A344775.

T(n,0) are the Bell numbers. T(2k,k) are the numbers of 2-balanced partitions in the particular case in which the third set is empty. T(2k,k) are the generalized Bell numbers given in A023998.

			T(4,1) = 5, number of 2-balanced partitions of a set A of 4 elements with 1 element in the first subset and 1 element in the second subset: A={a} U {b} U {c,d}. The five partitions are: ((a,b),(c),(d)), ((a,b),(c,d)), ((a,b,c),(d)), ((a,b,d),(c)), ((a,b,c,d)). Note that if a block contains a, then it must contain b. Thus, T(n,1) = T(n-1,0).
Triangle T(n,k) begins:
        1;
        1;
        2,      1;
        5,      2;
       15,      5,      3;
       52,     15,      8;
      203,     52,     25,    16;
      877,    203,     89,    53;
     4140,    877,    354,   197,   131;
    21147,   4140,   1551,   810,   512;
   115975,  21147,   7403,  3643,  2193,  1496;
   678570, 115975,  38154, 17759, 10201,  6697;
  4213597, 678570, 210803, 93130, 51146, 32345, 22482;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Francesca Aicardi, Balanced partitions, preprint on researchgate, 2021.
Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Brauer and Jones tied monoids, arXiv:2107.04170 [math.RT], 2021.

Cf. A000110 (Bell numbers), A023998, A061691 (generalized Stirling numbers), A344775 (row sums).

T(n,k) = Sum_{j=1..n-k} C(n,k,j). C(n,k,j) is defined for n>=2k, j<=n-k, and obtained by the recursion: C(n,k,j) = C(n-1,k,j-1) + j*C(n-1,k,j), with initial conditions C(2k,k,j) = triangle A061691(k,j) of generalized Stirling numbers.

A346220 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( (BesselI(0,2sqrt(x)) - BesselJ(0,2sqrt(x))) / 2 ).

Original entry on oeis.org

1, 1, 2, 7, 40, 321, 3356, 45123, 752256, 15018433, 355378732, 9823042923, 311510611072, 11242338245009, 458052976883672, 20851748359005567, 1054108827258438656, 58860837547461314049, 3606677286494115444812, 241397002229305033296603, 17579096976247770110062080

Offset: 0

Ilya Gutkovskiy, Jul 11 2021

nonn

Cf. A003724, A023998.

nmax = 20; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[x]] - BesselJ[0, 2 Sqrt[x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, 2 k + 1]^2 (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 20}]

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=0} x^(2*n+1) / ((2*n + 1)!)^2 ).

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

A336635 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^2 - 1).

Original entry on oeis.org

1, 2, 14, 176, 3470, 96792, 3590048, 169686792, 9903471502, 696692504552, 57958925154584, 5614276497440712, 625153195794408608, 79159558899671117896, 11293672011942106846808, 1801015209162807119535216, 318805481931592799427378062

Offset: 0

Ilya Gutkovskiy, Jul 28 2020

nonn

Cf. A000984, A023998, A055882, A323666, A336636, A336637.

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

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

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

Ilya Gutkovskiy, Jul 28 2020

nonn

Cf. A002893, A023998, A247452, A336635, A336637.

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}]

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

A336637 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^4 - 1).

Original entry on oeis.org

1, 4, 60, 1648, 71612, 4448384, 370135632, 39480942848, 5227020747708, 837878205997216, 159457868003008640, 35459969754432262208, 9093585253916177728592, 2659611377508767798535488, 878768601771275773332660736, 325350340926291926090183214848

Offset: 0

Ilya Gutkovskiy, Jul 28 2020

nonn

Cf. A002895, A023998, A336635, A336636.

nmax = 15; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]]^4 - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[2 k, k] HypergeometricPFQ[{1/2, -k, -k, -k}, {1, 1, 1/2 - k}, 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

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

A061697 Generalized Bell numbers.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 201, 1226, 5587, 493333, 8910253, 109739620, 6832444928, 251336859489, 6402632091649, 369288128260091, 21333939590516867, 941843896620169405, 60266201588496408645, 4623833509894543300868, 309412778502377193367456, 24102475277979402591991181

Offset: 0

N. J. A. Sloane, Jun 19 2001

nonn

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Cf. A023998, A061696.

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x)) - 1 - x - x^2/4). - Ilya Gutkovskiy, Jul 12 2020

More terms from Ilya Gutkovskiy, Jul 12 2020

A337826 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^4 * a(n-k).

Original entry on oeis.org

1, 1, 10, 105, 2248, 62445, 2390436, 116650177, 7043659904, 514744959321, 44534754680500, 4493090921151261, 521600149636044480, 68900819660071184149, 10259571068808850618480, 1708054303772376318547125, 315688007001129064574027776, 64370788231256983836207599153

Offset: 0

Ilya Gutkovskiy, Sep 24 2020

nonn

Cf. A023998, A279358, A336227, A337591, A337825.

a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k^4 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x (BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])], {x, 0, nmax}], x] Range[0, nmax]!^2

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

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} n^3 * x^n / (n!)^2).

A346271 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^2 / 4 ).

Original entry on oeis.org

1, 1, 2, 7, 41, 346, 3807, 53747, 952275, 20362552, 515112983, 15277888693, 523304644304, 20415373547609, 900219731675981, 44533809102813206, 2451041479421900803, 149140880201760643360, 9982798939295116151967, 731215136812226462200109, 58333374310397488522052976

Offset: 0

Ilya Gutkovskiy, Jul 12 2021

nonn

Cf. A023998, A061696, A097514, A346272.

nmax = 20; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]] - 1 - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = n a[n - 1] + (1/n) Sum[Binomial[n, k]^2 k a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 20}]

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + Sum_{n>=3} x^n / (n!)^2 ).

a(0) = 1; a(n) = n * a(n-1) + (1/n) * Sum_{k=3..n} binomial(n,k)^2 * k * a(n-k).

A346272 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^3 / 36 ).

Original entry on oeis.org

1, 1, 3, 15, 115, 1196, 16282, 276158, 5713507, 140482000, 4047179258, 134447125418, 5097852537802, 218254682152053, 10469861372693621, 558373926672949031, 32908746221003292003, 2130712239317226923136, 150826951188229240683858, 11618459541824256750732794

Offset: 0

Ilya Gutkovskiy, Jul 12 2021

nonn

Cf. A023998, A061696, A124504, A346271.

nmax = 19; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]] - 1 - x^3/36], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = a[1] = 1; a[n_] := a[n] = n a[n - 1] + n (n - 1)^2 a[n - 2]/2 + (1/n) Sum[Binomial[n, k]^2 k a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 19}]

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + x^2 / 4 + Sum_{n>=4} x^n / (n!)^2 ).

a(0) = a(1) = 1; a(n) = n * a(n-1) + n * (n-1)^2 * a(n-2) / 2 + (1/n) * Sum_{k=4..n} binomial(n,k)^2 * k * a(n-k).

cp's OEIS Frontend

A342196 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^2 * a(k-1).

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A343254 Triangle read by rows: T(n,k) is the number of 2-balanced partitions of a set of n elements in which the first and the second subsets have cardinality k, for n >= 0, k = 0..floor(n/2).

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A346220 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( (BesselI(0,2*sqrt(x)) - BesselJ(0,2*sqrt(x))) / 2 ).

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A336635 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^2 - 1).

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A336636 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^3 - 1).

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A336637 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^4 - 1).

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A061697 Generalized Bell numbers.

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A337826 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^4 * a(n-k).

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A346271 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^2 / 4 ).

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A346272 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^3 / 36 ).

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A346220 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( (BesselI(0,2sqrt(x)) - BesselJ(0,2sqrt(x))) / 2 ).