A323340 -id:A323340 - cp's OEIS frontend

A323339 Numerator of the sum of inverse products of parts in all compositions of n.

1, 1, 3, 7, 11, 347, 3289, 1011, 38371, 136553, 4320019, 12528587, 40771123, 29346499543, 129990006917, 1927874590951, 903657004321, 437445829053473, 12456509813711881, 187206004658210129, 1974369484466728177, 1967745662306280217, 21401375717067880189

Offset: 0

Alois P. Heinz, Jan 11 2019

Numerators of the INVERT transform of reciprocal integers.

			1/1, 1/1, 3/2, 7/3, 11/3, 347/60, 3289/360, 1011/70, 38371/1680, 136553/3780, 4320019/75600, 12528587/138600, 40771123/285120, ... = A323339/A323340

Alois P. Heinz, Table of n, a(n) for n = 0..466

Denominators: A323340.

Cf. A000142, A007840, A011782, A088305, A177208, A177209, A322364, A322365, A322380, A322381, A323290, A323291.

b:= proc(n) option remember;
     `if`(n=0, 1, add(b(n-j)/j, j=1..n))
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..25);

nmax = 20; Numerator[CoefficientList[Series[1/(1 + Log[1-x]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Feb 12 2024 *)

G.f. for fractions: 1 / (1 + log(1 - x)). - Ilya Gutkovskiy, Nov 12 2019
a(n) = numerator( A007840(n)/n! ). - Alois P. Heinz, Jan 04 2024
A323339(n)/A323340(n) ~ exp(n) / (exp(1) - 1)^(n+1). - Vaclav Kotesovec, Feb 12 2024

A177208 Numerators of exponential transform of 1/n.

Original entry on oeis.org

1, 1, 3, 17, 19, 81, 8351, 184553, 52907, 1768847, 70442753, 1096172081, 22198464713, 195894185831, 42653714271997, 30188596935106763, 20689743895700791, 670597992748852241, 71867806446352961329, 8445943795439038164379, 379371134635840861537

Offset: 0

Franklin T. Adams-Watters, May 04 2010

b(n) = a(n)/A177209(n) is the sum over all set partitions of [n] of the product of the reciprocals of the part sizes.

Numerators of moments of Dickman-De Bruijn distribution as shown on page 257 of Cellarosi and Sinai. [Jonathan Vos Post, Jan 07 2012]

			For n=4, there is 1 set partition with a single part of size 4, 4 with sizes [3,1], 3 with sizes [2,2], 6 with sizes [2,1,1], and 1 with sizes [1,1,1,1]; so b(4) = 1/4 + 4/(3*1) + 3/(2*2) + 6/(2*1*1) + 1/(1^4) = 1/4 + 4/3 + 3/4 + 3 + 1 = 19/4.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), pp. 228-230.
Knuth, Donald E., and Luis Trabb Pardo. "Analysis of a simple factorization algorithm." Theoretical Computer Science 3.3 (1976): 321-348. See Eq. (6.6) and (6.7), page 334.

Alois P. Heinz, Table of n, a(n) for n = 0..200
F. Cellarosi and Ya. G. Sinai, The Möbius function and statistical mechanics, Bull. Math. Sci., 2011.
Wikipedia, Exponential integral

Denominators are in A177209.

Cf. A000110, A000248, A001620, A322364, A322365, A322380, A322381, A323339, A323340.

b:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*b(n-j)/j, j=1..n))
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..25); # Alois P. Heinz, Jan 08 2012

b[n_] := b[n] = If[n==0, 1, Sum[Binomial[n-1, j-1]*b[n-j]/j, {j, 1, n}]]; a[n_] := Numerator[b[n]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)

Vec(serlaplace(exp(sum(n=1,30,x^n/(n*n!),O(x^31)))))

E.g.f. for fractions is exp(f(z)), where f(z) = sum(k>0, z^k/(k*k!)) = integral(0..z,(exp(t)-1)/t dt) = Ei(z) - gamma - log(z) = -Ein(-z). Here gamma is Euler's constant, and Ei and Ein are variants of the exponential integral.

Knuth & Trabb-Pardo (6.7) gives a recurrence. - N. J. A. Sloane, Nov 09 2022

A322364 Numerator of the sum of inverse products of parts in all partitions of n.

Original entry on oeis.org

1, 1, 3, 11, 7, 27, 581, 4583, 2327, 69761, 775643, 147941, 30601201, 30679433, 10928023, 6516099439, 445868889691, 298288331489, 7327135996801, 1029216937671847, 14361631943741, 837902013393451, 2766939485246012129, 274082602410356881, 835547516381094139939

Offset: 0

Alois P. Heinz, Dec 04 2018

			1/1, 1/1, 3/2, 11/6, 7/3, 27/10, 581/180, 4583/1260, 2327/560, 69761/15120, 775643/151200, 147941/26400, 30601201/4989600, 30679433/4633200 ... = A322364/A322365

Alois P. Heinz, Table of n, a(n) for n = 0..505
A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
D. Zeilberger, N. Zeilberger, Fractional Counting of Integer Partitions, 2018.

Denominators: A322365.

Cf. A000041, A006906, A080130, A177208, A177209, A322380, A322381, A323290, A323291, A323339, A323340.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +b(n-i, min(i, n-i))/i)
    end:
a:= n-> numer(b(n$2)):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n==0||i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]/i];
a[n_] := Numerator[b[n, n]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

a(n) = {my(s=0); forpart(p=n, s += 1/vecprod(Vec(p))); numerator(s);} \\ Michel Marcus, Apr 29 2020

Limit_{n-> infinity} a(n)/(n*A322365(n)) = exp(-gamma) = A080130.

A322380 Numerator of the sum of inverse products of parts in all strict partitions of n.

Original entry on oeis.org

1, 1, 1, 5, 7, 37, 79, 173, 101, 127, 1033, 1571, 200069, 2564519, 5126711, 25661369, 532393, 431100529, 1855391, 1533985991, 48977868113, 342880481117, 342289639579, 435979161889, 1308720597671, 373092965489, 7824703695283, 24141028973, 31250466692609

Offset: 0

Alois P. Heinz, Dec 05 2018

a(n)/A322381(n) = A007838(n)/A000142(n) is the probability that a random permutation of [n] has distinct cycle sizes. - Geoffrey Critzer, Feb 23 2022

			1/1, 1/1, 1/2, 5/6, 7/12, 37/60, 79/120, 173/280, 101/168, 127/210, 1033/1680, 1571/2640, 200069/332640, 2564519/4324320, 5126711/8648640, ... = A322380/A322381

Alois P. Heinz, Table of n, a(n) for n = 0..1268
Andreas B. G. Blobel, An Asymptotic Form of the Generating Function Prod_{k=1,oo} (1+x^k/k), arXiv:1904.07808 [math.CO], 2019.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 137.
A. Knopfmacher and J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
D. Zeilberger, N. Zeilberger, Fractional Counting of Integer Partitions, 2018.

Denominators: A322381.

Cf. A000009, A000142, A007838, A022629, A080130, A177208, A177209, A322364, A322365, A323290, A323291, A323339, A323340.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +b(n-i, min(i-1, n-i))/i))
    end:
a:= n-> numer(b(n$2)):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[i - 1, n - i]]/i]];
a[n_] := Numerator[b[n, n]];
a /@ Range[0, 30] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

Limit_{n->infinity} a(n)/A322381(n) = exp(-gamma) = A080130.

Sum_{n>=0} a(n)/A322381(n)*x^n = Product_{i>=1} (1 + x^i/i). - Geoffrey Critzer, Feb 23 2022

A323290 Numerator of the sum of inverse products of cycle sizes in all permutations of [n].

Original entry on oeis.org

1, 1, 3, 19, 107, 641, 51103, 1897879, 7860361, 505249081, 40865339743, 1355547261301, 244350418462637, 34907820791828741, 1949845703291363567, 1136592473036395958917, 31690844708764028510969, 2681369908698254192692979, 768531714669026186032238737

Offset: 0

Alois P. Heinz, Jan 09 2019

			1/1, 1/1, 3/2, 19/6, 107/12, 641/20, 51103/360, 1897879/2520, 7860361/1680, 505249081/15120, 40865339743/151200, ... = A323290/A323291

Alois P. Heinz, Table of n, a(n) for n = 0..270

Denominators: A323291.

Cf. A000142, A177208, A177209, A322364, A322365, A322380, A322381, A323339, A323340.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1)*(j-1)!/j, j=1..n))
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..20);

nmax = 20; Numerator[CoefficientList[Series[Exp[PolyLog[2, x]], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Feb 12 2024 *)

E.g.f.: exp(polylog(2,x)) (for fractions A323290(n)/A323291(n)). - Vaclav Kotesovec, Feb 12 2024

A323290(n)/A323291(n) ~ exp(Pi^2/6) * n! / n^2. - Vaclav Kotesovec, Feb 14 2024

A365594 The denominators of a series that converges to 1/e obtained using Whittaker's Root Series Formula.

Original entry on oeis.org

3, 42, 154, 3817, 1141283, 119706444, 1396550916, 20958700652, 2359646218028, 324742403298918, 107268957934572210, 41877140987048387615, 19073758392921536694655, 10024177256513161424322680, 376301673554116445531842536, 10673126660749797308728534491

Offset: 1

Raul Prisacariu, Sep 10 2023

The Whittaker's Root Series Formula is applied to 1+x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/6 +..., which is 1 + the Taylor expansion of log(1+x). The series obtained after applying Whittaker's Root Series Formula: 1/e-1=(-1)/1+(1/2)/(1*3/2)+(1/12)/((3/2)*(7/3))+(1/18)/((7/3)*(11/3))+(1/20)/((11/3)*(347/60))+(563/10800)/((347/60)*(3289/360))+ ... . The series can be simplified to: 1/e=1/3+1/42+1/154+9/3817+1126/1141283+ ... . The sequence is formed by the denominators of the simplified series.

The fractions in the denominators of the non-simplified series seem to be equal to terms from A323339 divided by the corresponding terms from A323340. Thus, the Whittaker's Root Series for 1 + the Taylor expansion of log(1+x) offers an alternative method for obtaining the terms of A323339 and A323340 using the determinants of Toeplitz matrices (formed using the coefficients of 1 + the Taylor expansion of log(1+x)).

			Whittaker's Root Series Formula is applied to 1 + the Taylor expansion of log(1+x) and the terms are simplified. The sequence is formed by the denominators of the simplified terms, starting with the second term in the Whittaker's Root Series.
a(1) is the denominator of -(-1/2)/(1*det((1,-1/2),(1,1))) = (1/2)/(3/2) = 1/3.
a(2) is the denominator of -det((-1/2,1/3),(1,-1/2))/(det((1,-1/2),(1,1))*det((1,-1/2,1/3),(1,1,-1/2),(0,1,1))) = (1/12)/((3/2)*(7/3)) = 1/42.

Raul Prisacariu, Whittaker's Root Series: Going Transcendental.
E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A068985 (1/e), A365595 (numerators), A323339, A323340.

c[k_] := If[k < 0, 0, SeriesCoefficient[1 + Log[1 + x], {x, 0, k}]]; Table[-Det[ToeplitzMatrix[Table[c[3 - j], {j, 1, n}], Table[c[j + 1], {j, 1, n}]]] / (Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n}], Table[c[j], {j, 1, n}]]] * Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n + 1}], Table[c[j], {j, 1, n + 1}]]]), {n, 1, 20}] // Denominator (* Vaclav Kotesovec, Oct 09 2023 *)

a(n) is the denominator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=1, c(2)=-1/2, c(3)=1/3, c(4)=-1/4, c(n)=(1/n)*(-1)^(n+1).c(n) is simply the coefficient of x^n in the series formed by 1+ the Taylor expansion of log(1+x).

a(6)-a(7) corrected and extended by Vaclav Kotesovec, Oct 09 2023

A365595 The numerators of a series that converges to 1/e obtained using Whittaker's Root Series Formula.

Original entry on oeis.org

1, 1, 1, 9, 1126, 53825, 302989, 2285199, 133296721, 9731109349, 1737376806937, 372236638394027, 94229801087550639, 27818002500902930641, 591930814558449521261, 9591188150350759241842, 2816408483135723327055984, 1394771058490469072473603553, 385768133102988434073147277769

Offset: 1

Raul Prisacariu, Sep 10 2023

The Whittaker's Root Series Formula is applied to 1+x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/6 +..., which is 1 + the Taylor expansion of log(1+x). The series obtained after applying Whittaker's Root Series Formula: 1/e-1=(-1)/1+(1/2)/(1*3/2)+(1/12)/((3/2)*(7/3))+(1/18)/((7/3)*(11/3))+(1/20)/((11/3)*(347/60))+(563/10800)/((347/60)*(3289/360))+ ... . The series can be simplified to: 1/e=1/3+1/42+1/154+9/3817+1126/1141283+ ... . The sequence is formed by the numerators of the simplified series.

The fractions in the denominators of the non-simplified series seem to be equal to terms from A323339 divided by the corresponding terms from A323340. Thus, the Whittaker's Root Series for 1 + the Taylor expansion of log(1+x) offers an alternative method for obtaining the terms of A323339 and A323340 using the determinants of Toeplitz matrices (formed using the coefficients of 1 + the Taylor expansion of log(1+x)).

			Whittaker's Root Series Formula is applied to 1 + the Taylor expansion of log(1+x) and the terms are simplified. The sequence is formed by the numerators of the simplified terms, starting with the second term in the Whittaker's Root Series.
a(1) is the numerator of -(-1/2)/(1*det((1,-1/2),(1,1))) = (1/2)/(3/2) = 1/3.
a(2) is the numerator of -det((-1/2,1/3),(1,-1/2))/(det((1,-1/2),(1,1))*det((1,-1/2,1/3),(1,1,-1/2),(0,1,1))) = (1/12)/((3/2)*(7/3)) = 1/42.

Raul Prisacariu, Whittaker's Root Series: Going Transcendental.
E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A068985 (1/e), A365594 (denominators), A323339, A323340.

c[k_] := If[k < 0, 0, SeriesCoefficient[1 + Log[1 + x], {x, 0, k}]]; Table[-Det[ToeplitzMatrix[Table[c[3 - j], {j, 1, n}], Table[c[j + 1], {j, 1, n}]]] / (Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n}], Table[c[j], {j, 1, n}]]] * Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n + 1}], Table[c[j], {j, 1, n + 1}]]]), {n, 1, 20}] // Numerator (* Vaclav Kotesovec, Oct 09 2023 *)

a(n) is the numerator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=1, c(2)=-1/2, c(3)=1/3, c(4)=-1/4, c(n)=(1/n)*(-1)^(n+1).c(n) is simply the coefficient of x^n in the series formed by 1+ the Taylor expansion of log(1+x).

More terms from Vaclav Kotesovec, Oct 09 2023

A368754 a(n) = (n!)^n * [x^n] * 1/(1 - polylog(n,x)).

Original entry on oeis.org

1, 1, 5, 278, 404768, 28436662624, 151309093659896512, 86745908552613198656020224, 7184659625769578063908866060107907072, 110866279942987479997999976181870531647691458347008, 399488258540989429698770032526869852804662313023226648081962369024

Offset: 0

Alois P. Heinz, Jan 04 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..30

Cf. A000051, A000142, A007840, A011782, A036740, A323339, A323340, A336258, A336259, A336260, A336261.

a:= n-> n!^n*coeff(series(1/(1-polylog(n, x)), x, n+1), x, n):
seq(a(n), n=0..10);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)/j^k, j=1..n))
    end:
a:= n-> n!^n*b(n$2):
seq(a(n), n=0..10);

a(n) = (n!)^n*b(n,n) with b(n,k) = Sum_{j=1..n} b(n-j,k)/j^k for n>0, b(0,k) = 1.

cp's OEIS Frontend

A323339 Numerator of the sum of inverse products of parts in all compositions of n.

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A177208 Numerators of exponential transform of 1/n.

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A322364 Numerator of the sum of inverse products of parts in all partitions of n.

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A322380 Numerator of the sum of inverse products of parts in all strict partitions of n.

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A323290 Numerator of the sum of inverse products of cycle sizes in all permutations of [n].

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A365594 The denominators of a series that converges to 1/e obtained using Whittaker's Root Series Formula.

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A365595 The numerators of a series that converges to 1/e obtained using Whittaker's Root Series Formula.

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