A053557 -id:A053557 - cp's OEIS frontend

A053519 Denominators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))).

1, 3, 15, 29, 597, 4701, 4643, 413691, 4512993, 17926611, 695000919, 9680369943, 4380611853, 2303928046437, 39031251610227, 25940523189513, 1206420504316107, 20365306128628437, 1849040492948486661

Offset: 0

N. J. A. Sloane, Jan 15 2000

Also numerators of successive convergents to continued fraction 1/(2+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/9+...))))))).

A053518/A053519 -> (2*e-5)/(3-e) = 1.5496467783... as n-> infinity.

			Convergents (to the first continued fraction) are 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.

Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748.
L. Euler, Introduction à l'analyse infinitésimale, Tome premier, Tome second, trad. du latin en français par J. B. Labey, Paris, 1796-1797.
M. A. Stern, Theorie der Kettenbrüche und ihre Anwendung, Crelle, 1832, pp. 1-22.

Cf. A053518, A053520, A053556, A053557.

for j from 1 to 50 do printf(`%d,`,denom(cfrac([1,seq([i,i+1],i=2..j)]))); od:

num[0]=1; num[1]=5; num[n_] := num[n] = (n+2)*num[n-1] + (n+1)*num[n-2]; den[0]=1; den[1]=3; den[n_] := den[n] = (n+2)*den[n-1] + (n+1)*den[n-2]; a[n_] := Denominator[num[n]/den[n]]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jan 16 2013 *)

Thanks to R. K. Guy, Steven Finch, Bill Gosper for comments

More terms from James Sellers, Feb 02 2000

A103360 Denominator of coefficient in the interpolation polynomial for initial values of the factorial, read by row.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 2, 6, 1, 8, 12, 8, 12, 1, 30, 24, 12, 24, 60, 1, 144, 240, 144, 48, 36, 20, 1, 280, 720, 240, 144, 240, 180, 140, 1, 5760, 10080, 960, 180, 640, 1440, 1440, 280, 1

Offset: 0

Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Feb 02 2005

N(n,n)/D(n,n) = Sum{k=0..n}(-1)^k/k! = A000166/n! = A053557/A053556, where N(n,n) is A103361.

			1; 1; 1/2*x^2-1/2*x+1; 1/3*x^3-1/2*x^2+1/6*x+1;
3/8*x^4-23/12*x^3+29/8*x^2-25/12*x+1;
11/30*x^5-79/24*x^4+131/12*x^3-353/24*x^2+403/60*x+1

Cf. A103361, A000166, A053556, A053557.

D(n, k) in Sum{k=0..n}N(n, k)/D(n, k)*m^k=m!, m=0..n, with reduced fraction N(n, k)/D(n, k) and N(n, k) is A103361. D(n, k)=a(n*(n+3)/2-k).

A103361 Numerator of coefficient in the interpolation polynomial for initial values of the factorial, read by rows.

Original entry on oeis.org

1, 0, 1, 1, -1, 1, 1, -1, 1, 1, 3, -23, 29, -25, 1, 11, -79, 131, -353, 403, 1, 53, -1237, 4031, -3451, 3101, -749, 1, 103, -5297, 14213, -34903, 126121, -101297, 31837, 1, 2199, -100123, 106657, -119129, 1438599, -6199951, 6112397, -455481, 1

Offset: 0

Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Feb 02 2005

Denominator N(n,n) of leading coefficient of the n-th polynomial = n-th term of A053557. N(n,n)/D(n,n) = Sum{k=0..n}(-1)^k/k! = A000166/n! = A053557/A053556, where D(n,n) is A103360.

			1; 1; 1/2*x^2-1/2*x+1; 1/3*x^3-1/2*x^2+1/6*x+1;
3/8*x^4-23/12*x^3+29/8*x^2-25/12*x+1;
11/30*x^5-79/24*x^4+131/12*x^3-353/24*x^2+403/60*x+1

Cf. A103360, A000166, A053556, A053557.

N(n, k) in Sum{k=0..n}N(n, k)/D(n, k)*m^k=m!, m=0..n, with reduced fraction N(n, k)/D(n, k) and D(n, k) is A103360. N(n, k)=a(n*(n+3)/2-k).

A354298 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

Original entry on oeis.org

1, 2, 11, 76, 137, 7534, 97943, 1469144, 24975449, 94906706, 9965204131, 229199695012, 5729992375301, 9100576125478, 897316805972131, 563093542209232, 4589775462547450033, 5539384178936577626, 5943759223998947792699, 46361321947191792783052, 9504070999174317520525661

Offset: 1

Ilya Gutkovskiy, May 23 2022

			1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...

Robert Israel, Table of n, a(n) for n = 1..403

Cf. A001147, A053557, A061354, A064646, A103816, A113012, A120265, A143382, A289381, A306858, A354299 (denominators).

S:= 0: R:= NULL:
for n from 1 to 100 do
  S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
  R:= R, numer(S);
od:
R; # Robert Israel, Jan 10 2024

Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Numerator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Numerator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Numerator

Numerators of coefficients in expansion of sqrt(Pi*x*exp(-x)/2) * erfi(sqrt(x/2)) / (1 - x).

A353545 a(n) is the numerator of Sum_{k=1..n} 1 / (k*k!).

Original entry on oeis.org

1, 5, 47, 379, 9487, 14233, 87179, 44635753, 1205165611, 6025828181, 729125211161, 972166948343, 54765404757169, 71879593743829, 25876653747779441, 6624423359431551911, 1914458350875718742519, 51690375473644406388353, 18660225545985630712321553, 186602255459856307126125437

Offset: 1

Ilya Gutkovskiy, May 25 2022

			1, 5/4, 47/36, 379/288, 9487/7200, 14233/10800, 87179/66150, ...

Cf. A001563, A053557, A061354, A103816, A120265, A229837, A354401 (denominators), A354402.

Table[Sum[1/(k k!), {k, 1, n}], {n, 1, 20}] // Numerator
nmax = 20; Assuming[x > 0, CoefficientList[Series[(ExpIntegralEi[x] - Log[x] - EulerGamma)/(1 - x), {x, 0, nmax}], x]] // Numerator // Rest

a(n) = numerator(sum(k=1, n, 1/(k*k!))); \\ Michel Marcus, May 26 2022

from math import factorial
from fractions import Fraction
def A353545(n): return sum(Fraction(1, k*factorial(k)) for k in range(1,n+1)).numerator # Chai Wah Wu, May 27 2022

Numerators of coefficients in expansion of (Ei(x) - log(x) - gamma) / (1 - x), x > 0.

A354402 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).

Original entry on oeis.org

1, 3, 29, 229, 5737, 8603, 210781, 26979863, 728456581, 3642282779, 440716217519, 1762864869691, 297924162982399, 260683642609331, 15641018556560861, 4004100750479565401, 1157185116888594641129, 31243998155992054970143, 11279083334313131850347743, 112790833343131318500567523

Offset: 1

Ilya Gutkovskiy, May 25 2022

			1, 3/4, 29/36, 229/288, 5737/7200, 8603/10800, 210781/264600, ...

Cf. A001563, A053557, A061354, A103816, A120265, A239069, A353545, A354404 (denominators).

Table[Sum[(-1)^(k + 1)/(k k!), {k, 1, n}], {n, 1, 20}] // Numerator
nmax = 20; Assuming[x > 0, CoefficientList[Series[(EulerGamma + Log[x] - ExpIntegralEi[-x])/(1 - x), {x, 0, nmax}], x]] // Numerator // Rest

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/(k*k!))); \\ Michel Marcus, May 26 2022

from math import factorial
from fractions import Fraction
def A354402(n): return sum(Fraction(1 if k & 1 else -1, k*factorial(k)) for k in range(1,n+1)).numerator # Chai Wah Wu, May 27 2022

Numerators of coefficients in expansion of (gamma + log(x) - Ei(-x)) / (1 - x), x > 0.

A373417 Triangle T(n,k) for the number of permutations in symmetric group S_n with (n-k) fixed points and an even number of non-fixed point cycles. Equivalent to the number of cycles of n items with cycle type defined by non-unity partitions of k<=n that contain an even number of parts.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 15, 20, 1, 0, 0, 0, 45, 120, 130, 1, 0, 0, 0, 105, 420, 910, 924, 1, 0, 0, 0, 210, 1120, 3640, 7392, 7413, 1, 0, 0, 0, 378, 2520, 10920, 33264, 66717, 66744, 1, 0, 0, 0, 630, 5040, 27300, 110880, 333585, 667440, 667476

Offset: 0

Julian Hatfield Iacoponi, Jun 04 2024

A343418(n) + a(n) = A098825(n) = partial derangement "rencontres" triangle.
A343418(n) - a(n) = (k-1) * binomial(n,k) = A127717(n-1,k-1).
Difference of 1st and 2nd leading diagonals (n > 0).
T(n,n) - T(n,n-1) = -1,0,0,3,5,10,14,21,27,36,44,...
                  = (-1) + (1+0) + (3+2) + (5+4) + (7+6) + (9+8) + ...
Cf. A176222(n) with 2 terms -1,0 prepended (moving its offset from 3 to 1).

			Triangle array T(n,k):
  n:  {k<=n}
  0:  {1}
  1:  {1, 0}
  2:  {1, 0, 0}
  3:  {1, 0, 0, 0}
  4:  {1, 0, 0, 0,   3}
  5:  {1, 0, 0, 0,  15,   20}
  6:  {1, 0, 0, 0,  45,  120,   130}
  7:  {1, 0, 0, 0, 105,  420,   910,    924}
  8:  {1, 0, 0, 0, 210, 1120,  3640,   7392,   7413}
  9:  {1, 0, 0, 0, 378, 2520, 10920,  33264,  66717,  66744}
  10: {1, 0, 0, 0, 630, 5040, 27300, 110880, 333585, 667440, 667476}
T(n,0) = 1 due to sole permutation in S_n with n fixed points, namely the identity permutation, with 0 non-fixed point cycles, an even number. (T(0,0)=1 relies on S_0 containing an empty permutation.)
T(n,1) = 0 due to no permutations in S_n with (n-1) fixed points.
T(n,2) = T(n,3) = 0 due to only non-unity partitions of 2 and 3 being of odd length, namely the trivial partitions (2),(3).
Example:
T(4,4) = 3 since S_4 contains 3 permutations with 0 fixed points and an even number of non-fixed point cycles, namely the derangements: (12)(34),(13)(24),(14)(23).
Worked Example:
T(7,6) = 910 permutations in S_7 with 1 fixed point and an even number of non-fixed point cycles.
T(7,6) = 910 possible (4,2)- and (3,3)-cycles of 7 items.
N(n,y) = possible y-cycles of n items.
N(n,y) = (n!/(n-k)!) / (M(y) * s(y)).
y = partition of k<=n with q parts = (p_1, p_2, ..., p_i, ..., p_q)
s.t. k = Sum_{i=1..q} p_i.
Or:
y = partition of k<=n with d distinct parts, each with multiplicity m_j = (y_1^m_1, y_2^m_2, ..., y_j^m_j, ..., y_d^m_d)
s.t. k = Sum_{j=1..d} m_j*y_j.
M(y) = Product_{i=1..q} p_i = Product_{j=1..d} y_j^m_j.
s(y) = Product_{j=1..d} m_j! ("symmetry of repeated parts").
Note: (n!/(n-k)!) / s(y) = multinomial(n, {m_j}).
Therefore:
T(7,6) = N(7,y=(4,2)) + N(7,y=(3^2))
       = (7!/(4*2)) + (7!/(3^2)/2!)
       = 7! * (1/8 + 1/18)
       = 5040 * (13/72)
T(7,6) = 910.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened)

Cf. A373418 (odd case), A373339 (row sums), A216778 (main diagonal).

b:= proc(n, t) option remember; `if`(n=0, t, add(expand(`if`(j>1, x^j, 1)*
      b(n-j, irem(t+signum(j-1), 2)))*binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 04 2024

Table[Table[n!/(n-k)!/2 * (Sum[(-1)^j/j!, {j, 0, k}] - ((k - 1)/k!)), {k, 0, n}], {n, 0, 10}]

T(n,k) = (n!/(n-k)!/2) * (Sum_{j=0..k} (-1)^j/j! - (k-1)/k!) Cf. Sum_{j=0..k} (-1)^j/j! = A053557(k) / A053556(k).

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).

A354304 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (k!)^2.

Original entry on oeis.org

1, 0, 1, 2, 43, 403, 23213, 118483, 51997111, 1842647621, 327581799289, 8918414485643, 4670006130663971, 361730891537680087, 130890931830249779173, 427294615628884602769, 6534075316966068976316143, 885163015595247156635327497, 41526561745210509140249210357

Offset: 0

Ilya Gutkovskiy, May 23 2022

			1, 0, 1/4, 2/9, 43/192, 403/1800, 23213/103680, 118483/529200, 51997111/232243200, 1842647621/8230118400, ...

Cf. A001044, A053557, A061354, A073701, A091681, A103816, A120265, A143382, A354302, A354305 (denominators).

Table[Sum[(-1)^k/(k!)^2, {k, 0, n}], {n, 0, 18}] // Numerator
nmax = 18; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator

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

A373418 Triangle read by rows: T(n,k) is the number of permutations in symmetric group S_n with (n-k) fixed points and an odd number of non-fixed point cycles. Equivalent to the number of cycles of n items with cycle type defined by non-unity partitions of k <= n that contain an odd number of parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 3, 2, 0, 0, 6, 8, 6, 0, 0, 10, 20, 30, 24, 0, 0, 15, 40, 90, 144, 135, 0, 0, 21, 70, 210, 504, 945, 930, 0, 0, 28, 112, 420, 1344, 3780, 7440, 7420, 0, 0, 36, 168, 756, 3024, 11340, 33480, 66780, 66752, 0, 0, 45, 240, 1260, 6048, 28350, 111600, 333900, 667520, 667485

Offset: 0

Julian Hatfield Iacoponi, Jun 04 2024

a(n) + A343417(n) = A098825(n) = partial derangement "rencontres" triangle.
a(n) - A343417(n) = (k-1) * binomial(n,k) = A127717(n-1,k-1).
Difference of 2nd and 1st leading diagonals (n > 0):
T(n,n-1) - T(n,n) = 0,-1,1,2,6,9,15,20,28,35,45,54,...
                  = (0-1) + (2+1) + (4+3) + (6+5) + (8+7) + (10+9) + ...
Cf. A084265(n) with 2 terms 0,-1 prepended (moving its offset from 0 to -2).

			Triangle begins:
   n: {k<=n}
   0: {0}
   1: {0, 0}
   2: {0, 0,  1}
   3: {0, 0,  3,   2}
   4: {0, 0,  6,   8,    6}
   5: {0, 0, 10,  20,   30,   24}
   6: {0, 0, 15,  40,   90,  144,   135}
   7: {0, 0, 21,  70,  210,  504,   945,    930}
   8: {0, 0, 28, 112,  420, 1344,  3780,   7440,   7420}
   9: {0, 0, 36, 168,  756, 3024, 11340,  33480,  66780,  66752}
  10: {0, 0, 45, 240, 1260, 6048, 28350, 111600, 333900, 667520, 667485}
T(n,0) = 0 because the sole permutation in S_n with n fixed points, namely the identity permutation, has 0 non-fixed point cycles, not an odd number.
T(n,1) = 0 because there are no permutations in S_n with (n-1) fixed points.
Example:
T(3,3) = 2 since S_3 contains 3 permutations with 0 fixed points and an odd number of non-fixed point cycles, namely the derangements (123) and (132).
Worked Example:
T(7,6) = 945 permutations in S_7 with 1 fixed point and an odd number of non-fixed point cycles;
T(7,6) = 945 possible 6- and (2,2,2)-cycles of 7 items.
N(n,y) = possible y-cycles of n items;
N(n,y) = (n!/(n-k)!) / (M(y) * s(y)).
y = partition of k<=n with q parts = (p_1, p_2, ..., p_i, ..., p_q) such that k = Sum_{i=1..q} p_i.
Or:
y = partition of k<=n with d distinct parts, each with multiplicity m_j = (y_1^m_1, y_2^m_2, ..., y_j^m_j, ..., y_d^m_d) such that k = Sum_{j=1..d} m_j*y_j.
M(y) = Product_{i=1..q} p_i = Product_{j=1..d} y_j^m_j.
s(y) = Product_{j=1..d} m_j! ("symmetry of repeated parts").
Note: (n!/(n-k)!) / s(y) = multinomial(n, {m_j}).
Therefore:
T(7,6) = N(7,y=(6)) + N(7,y=(2^3))
       = (7!/6) + (7!/(2^3)/3!)
       = 7! * (1/6 + 1/48)
       = 5040 * (3/16);
T(7,6) = 945.

Cf. A373417 (even case), A373340 (row sums), A216779 (main diagonal).

b:= proc(n, t) option remember; `if`(n=0, t, add(expand(`if`(j>1, x^j, 1)*
      b(n-j, irem(t+signum(j-1), 2)))*binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);

Table[Table[n!/(n-k)!/2 * (Sum[(-1)^j/j!, {j, 0, k}] - ((k - 1)/k!)),{k,1,n}], {n,1,10}]

T(n,k) = (n!/(n-k)!/2) * ((Sum_{j=0..k} (-1)^j/j!) + (k-1)/k!). Cf. Sum_{j=0..k} (-1)^j/j! = A053557(k) / A053556(k).

cp's OEIS Frontend

A053519 Denominators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))).

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A103360 Denominator of coefficient in the interpolation polynomial for initial values of the factorial, read by row.

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A103361 Numerator of coefficient in the interpolation polynomial for initial values of the factorial, read by rows.

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A354298 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

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A353545 a(n) is the numerator of Sum_{k=1..n} 1 / (k*k!).

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A354402 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).

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A373417 Triangle T(n,k) for the number of permutations in symmetric group S_n with (n-k) fixed points and an even number of non-fixed point cycles. Equivalent to the number of cycles of n items with cycle type defined by non-unity partitions of k<=n that contain an even number of parts.

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

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