A006040 -id:A006040 - cp's OEIS frontend

A336807 a(n) = (n!)^2 * Sum_{k=0..n} 4^(n-k) / (k!)^2.

1, 5, 81, 2917, 186689, 18668901, 2688321745, 526911062021, 134889231877377, 43704111128270149, 17481644451308059601, 8461115914433100846885, 4873602766713466087805761, 3294555470298303075356694437, 2582931488713869611079648438609, 2324638339842482649971683594748101

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A006040, A056545, A336804, A336805, A336808.

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

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

a(0) = 1; a(n) = 4 * n^2 * a(n-1) + 1.

A336808 a(n) = (n!)^2 * Sum_{k=0..n} 5^(n-k) / (k!)^2.

Original entry on oeis.org

1, 6, 121, 5446, 435681, 54460126, 9802822681, 2401691556846, 768541298190721, 311259225767242006, 155629612883621003001, 94155915794590706815606, 67792259372105308907236321, 57284459169428986026614691246, 56138769986040406306082397421081, 63156116234295457094342697098716126

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A006040, A056546, A336804, A336805, A336807.

Table[n!^2 Sum[5^(n - k)/k!^2, {k, 0, n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - 5 x), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - 5*x).

a(0) = 1; a(n) = 5 * n^2 * a(n-1) + 1.

A180255 a(n) = n^2 * a(n-1) + n, a(0)=0.

Original entry on oeis.org

0, 1, 6, 57, 916, 22905, 824586, 40404721, 2585902152, 209458074321, 20945807432110, 2534442699285321, 364959748697086236, 61678197529807573897, 12088926715842284483826, 2720008511064514008860865, 696322178832515586268381456, 201237109682597004431562240801

Offset: 0

Groux Roland, Jan 17 2011

nonn

Integral_{x=0..1} x^n*BesselI(0,2*x^(1/2)) dx = A006040(n)*BesselI(1,2) - a(n)*BesselI(0,2). An elementary consequence is the irrationality of BesselI(0,2)/BesselI(1,2).

Paolo Xausa, Table of n, a(n) for n = 0..250 (terms 0..41 from Vincenzo Librandi)

Cf. A006040, A007526, A066998, A228229.

FoldList[#2^2*# + #2 &, Range[0, 20]] (* Paolo Xausa, Jun 19 2025 *)

a[0]:0$ a[n]:=n^2*a[n-1]+n$ makelist(a[n], n, 0, 15); /* Bruno Berselli, May 23 2011 */

a(n)=if(n==0,0,(n)^2*a(n-1)+(n));
for(n=0,12,print1(a(n),", "));  /* show terms */

From Seiichi Manyama, Jan 05 2024: (Start)
a(n) = (n!)^2 * Sum_{k=0..n} k/(k!)^2.
a(n) = n * A228229(n-1) for n > 0. (End)

A188808 T(n,k)=Number of nXk array permutations with each element remaining in its original row or its original column.

Original entry on oeis.org

1, 2, 2, 6, 9, 6, 24, 82, 82, 24, 120, 1313, 2720, 1313, 120, 720, 32826, 194568, 194568, 32826, 720, 5040, 1181737, 26101232, 72104097, 26101232, 1181737, 5040, 40320, 57905114, 5919004912, 57951767544, 57951767544, 5919004912, 57905114, 40320

Offset: 1

R. H. Hardin Apr 11 2011

Table starts
.....1........2..........6..........24.........120........720..........5040
.....2........9.........82........1313.......32826....1181737......57905114
.....6.......82.......2720......194568....26101232.5919004912.2103543163584
....24.....1313.....194568....72104097.57951767544
...120....32826...26101232.57951767544
...720..1181737.5919004912
..5040.57905114
.40320

			Some solutions for 4X3
..3..2.11....3..0..8....0.10..8....3..0..1....0.10.11....0..4..8....1..2..0
..9..1..4....4..1..5....6..7..3....9..7.11....6..5..4....3..1..5....6..4..5
..6..8..7....7..6.11....9..4..5....8.10..5....9..8..7....6..7..2....9..7..8
..0.10..5...10..9..2...11..1..2....6..4..2....3..1..2...11.10..9....3.10.11

R. H. Hardin, Table of n, a(n) for n = 1..39

Column 1 is A000142

Column 2 is A006040(n+1)

A193563 a(0) = 0, a(n) = n^2 * (a(n-1) + 1).

Original entry on oeis.org

0, 1, 8, 81, 1312, 32825, 1181736, 57905113, 3705927296, 300180111057, 30018011105800, 3632179343801921, 523033825507476768, 88392716510763573961, 17324972436109660496552, 3898118798124673611724425, 997918412319916444601453056

Offset: 0

Meherzad Lahewala, Jul 31 2011

Paolo Xausa, Table of n, a(n) for n = 0..250

Cf. A006040, A007526 (multiply by n instead of n^2), A180255.

seq(n!^2*add(1/k!^2,k=0..n-1),n=0..16);   # Mark van Hoeij, May 13 2013

FoldList[#2^2*(# + 1) &, Range[0, 20]] (* Paolo Xausa, Jun 18 2025 *)

a=[0];for(n=1,20,a=concat(a,(a[#a]+1)*n^2));a \\ Charles R Greathouse IV, Jul 31 2011

From Seiichi Manyama, Jan 05 2024: (Start)
a(n) = (n!)^2 * Sum_{k=0..n} (k/k!)^2.
a(n) = n^2 * A006040(n-1) for n > 0. (End)
a(n) = Sum_{k=1..n} (k!*binomial(n,k))^2. - Ridouane Oudra, Jun 14 2025
a(n) = n^2 + BesselI(0,2)*(n!)^2 - n^2*hypergeom([1], [n, n], 1) for n > 0. - Stefano Spezia, Jun 14 2025

A228230 Recurrence a(n) = (1/2)n(n+1)*a(n-1) + 1 with a(0) = 1.

Original entry on oeis.org

1, 2, 7, 43, 431, 6466, 135787, 3802037, 136873333, 6159299986, 338761499231, 22358258949247, 1743944198041267, 158698922021755298, 16663386812284306291, 1999606417474116754921, 271946472776479878669257, 41607810334801421436396322, 7114935567251043065623771063

Offset: 0

Peter Bala, Aug 19 2013

Cf. A006040 and A228229.

Seiichi Manyama, Table of n, a(n) for n = 0..269
E. W. Weisstein, Modified Bessel Function of the First Kind

Cf. A006040, A228229.

#A228230
a:=proc(n) option remember
if n = 0 then 1 else 1/2n(n+1)thisproc(n-1) + 1
fi
end:
seq(a(n), n = 0..20);

a(n) = (1/2^n)*n!*(n + 1)!*Sum_{k = 0..n} 2^k/(k!*(k + 1)!).
a(n) = n!*(n+1)!*(the coefficient of x^n*y^(n+1) in the expansion of exp(x + y)/(1 - x*y/2)).
G.f.: (1/(1 - x/2))*(1/sqrt(x))*BesselI(1, 2*sqrt(x)) = Sum_{n >= 0} a(n)*x^n/(n!*(n + 1)!).
Defining recurrence equation: a(n) = (1/2)*n*(n + 1)*a(n-1) + 1 with a(0) = 1.
Alternative recurrence equation: a(0) = 1, a(1) = 2, and for n >= 2, a(n) = ((1/2)*n*(n+1) + 1)*a(n-1) - (1/2)*n*(n - 1)*a(n-2).
The sequence b(n) := (1/2^n)*n!*(n + 1)! satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 1. It follows that we have the finite continued fraction expansion a(n) = (1/2^n)* n!*(n + 1)!*(1 + 1/(1 - 1/(4 - 3/(7 - ... - 1/2*n*(n - 1)/(1/2*n*(n + 1) + 1))))). Taking the limit yields the continued fraction expansion (1/sqrt(2))*BesselI(1,2*sqrt(2)) = Sum_{k >= 0} 2^k/(k!*(k + 1)!) = 1 + 1/(1 - 1/(4 - 3/(7 - 6/(11 - ... - (1/2)*n*(n - 1)/((1/2)*n*(n + 1) + 1 - ...))))) = 2.394833097....

Typo in the first formula corrected by Vaclav Kotesovec, Jul 02 2015

A336291 a(n) = (n!)^2 * Sum_{k=1..n} 1 / (k * ((n-k)!)^2).

Original entry on oeis.org

0, 1, 6, 39, 424, 7905, 227766, 9324511, 512970144, 36452217969, 3247711402870, 354391640998791, 46474986465907176, 7210874466760191409, 1306387103147257800774, 273269900360634449732895, 65363179181419926246184576, 17726298367452515070739268001

Offset: 0

Ilya Gutkovskiy, Jul 16 2020

nonn

Cf. A002104, A002720, A006040, A336292.

Table[(n!)^2 Sum[1/(k ((n - k)!)^2), {k, 1, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[-Log[1 - x] BesselI[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = (n!)^2 * sum(k=1, n, 1 / (k * ((n-k)!)^2)); \\ Michel Marcus, Jul 17 2020

Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(1 - x) * BesselI(0,2*sqrt(x)).
a(n) ~ BesselI(0,2) * (n!)^2 / n. - Vaclav Kotesovec, Jul 17 2020
a(n) = Sum_{k=1..n} (k-1)!*k!*binomial(n,k)^2. - Ridouane Oudra, Jun 15 2025

A343863 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (n!)^k * Sum_{j=1..n} (1/j!)^k.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 9, 16, 5, 1, 2, 17, 82, 65, 6, 1, 2, 33, 460, 1313, 326, 7, 1, 2, 65, 2674, 29441, 32826, 1957, 8, 1, 2, 129, 15796, 684545, 3680126, 1181737, 13700, 9, 1, 2, 257, 94042, 16175105, 427840626, 794907217, 57905114, 109601, 10

Offset: 0

Seiichi Manyama, May 02 2021

			Square array begins:
  1,   1,     1,       1,         1,           1, ...
  2,   2,     2,       2,         2,           2, ...
  3,   5,     9,      17,        33,          65, ...
  4,  16,    82,     460,      2674,       15796, ...
  5,  65,  1313,   29441,    684545,    16175105, ...
  6, 326, 32826, 3680126, 427840626, 50547203126, ...

Seiichi Manyama, Antidiagonals n = 0..59, flattened

Columns 0..3 give A000027(n+1), A000522, A006040, A217284.
Main diagonal gives A336247.
Cf. A291556.

T[n_, k_] := Sum[(n!/j!)^k, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)

PARI
```
T(n, k) = sum(j=0, n, (n!/j!)^k);
```

T(0,k) = 1 and T(n,k) = n^k * T(n-1,k) + 1 for n > 0.

A354302 a(n) is the numerator of Sum_{k=0..n} 1 / (k!)^2.

Original entry on oeis.org

1, 2, 9, 41, 1313, 5471, 1181737, 28952557, 1235309099, 150090055529, 30018011105801, 201787741322329, 523033825507476769, 44196358255381786981, 5774990812036553498851, 1949059399062336805862213, 997918412319916444601453057, 3697415655903280160125896583

Offset: 0

Ilya Gutkovskiy, May 23 2022

			1, 2, 9/4, 41/18, 1313/576, 5471/2400, 1181737/518400, 28952557/12700800, 1235309099/541900800, ...

Cf. A001044, A006040, A053557, A061354, A070910, A103816, A120265, A143382, A354303 (denominators), A354304.

Table[Sum[1/(k!)^2, {k, 0, n}], {n, 0, 17}] // Numerator
nmax = 17; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator

Numerators of coefficients in expansion of BesselI(0,2*sqrt(x)) / (1 - x).

A354303 a(n) is the denominator of Sum_{k=0..n} 1 / (k!)^2.

Original entry on oeis.org

1, 1, 4, 18, 576, 2400, 518400, 12700800, 541900800, 65840947200, 13168189440000, 88519495680000, 229442532802560000, 19387894021816320000, 2533351485517332480000, 855006126362099712000000, 437763136697395052544000000, 1621968544942912438272000000

Offset: 0

Ilya Gutkovskiy, May 23 2022

			1, 2, 9/4, 41/18, 1313/576, 5471/2400, 1181737/518400, 28952557/12700800, 1235309099/541900800, ...

Cf. A001044, A006040, A053556, A061355, A070910, A143383, A354302 (numerators), A354305.

Table[Sum[1/(k!)^2, {k, 0, n}], {n, 0, 17}] // Denominator
nmax = 17; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator

Denominators of coefficients in expansion of BesselI(0,2*sqrt(x)) / (1 - x).

cp's OEIS Frontend

A336807 a(n) = (n!)^2 * Sum_{k=0..n} 4^(n-k) / (k!)^2.

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A336808 a(n) = (n!)^2 * Sum_{k=0..n} 5^(n-k) / (k!)^2.

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A180255 a(n) = n^2 * a(n-1) + n, a(0)=0.

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A188808 T(n,k)=Number of nXk array permutations with each element remaining in its original row or its original column.

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A193563 a(0) = 0, a(n) = n^2 * (a(n-1) + 1).

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A228230 Recurrence a(n) = (1/2)*n*(n+1)*a(n-1) + 1 with a(0) = 1.

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A336291 a(n) = (n!)^2 * Sum_{k=1..n} 1 / (k * ((n-k)!)^2).

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A343863 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (n!)^k * Sum_{j=1..n} (1/j!)^k.

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A354302 a(n) is the numerator of Sum_{k=0..n} 1 / (k!)^2.

A228230 Recurrence a(n) = (1/2)n(n+1)*a(n-1) + 1 with a(0) = 1.