Christian Krattenthaler - cp's OEIS frontend

A131658 For n >= 1, put A_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j and B_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j * (Sum__{k=j+1..jn} (1/k)), and let u(n) be the largest integer for which exp(B_n(z)/(u(n)A_n(z))) has integral coefficients. The sequence is u(n).

1, 1, 1, 2, 2, 36, 36, 144, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, 29030400, 1567641600, 1567641600, 156764160000, 49380710400000, 217275125760000, 1086375628800000, 1738201006080000

Offset: 1

Christian Krattenthaler (Christian.Krattenthaler(AT)univie.ac.at), Sep 12 2007, Sep 30 2007

nonn

Different from A131657 and A056612.

Christian Krattenthaler, Table of n, a(n) for n = 1..40
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, 3(3) (2009), 555-591. [Part II appeared before Part I.]
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, Duke Math. J. 151(2) (2010), 175-218.

Cf. A007757 (bisection at even integers), A056612, A131657.

A formula, conditional on a widely believed conjecture, can be found in the article by Krattenthaler and Rivoal (2007-2009) cited in the references: see Theorem 4 and the accompanying remarks.

A131657 For n >= 1, put A_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j and B_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j * (Sum_{k=1..jn} (1/k)), and let b(n) be the largest integer for which exp(B_n(z)/(b(n)A_n(z))) has integral coefficients. The sequence is b(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 36, 36, 144, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, 29030400, 1567641600, 1567641600, 783820800000, 9876142080000, 651825377280000, 217275125760000, 8691005030400000

Offset: 1

Christian Krattenthaler (Christian.Krattenthaler(AT)univie.ac.at), Sep 12 2007, Sep 30 2007

nonn

Different from A131658 and A056612. The first difference between A056612 and this sequence occurs for n = 20, while the first difference between A056612 and A131658 occurs for n = 21.

Krattenhaler and Rivoal (2007-2009) conjecture that a(n) = n!*Xi(n), where Xi(1) = 1, Xi(7) = 1/140, and Xi(n) = Product_{p <= n} p^min(2, v_p(H_n)) for n <> 1, 7, where v_p(r) is the p-adic valuation of rational r. (Here p indicates a prime and H_n is the n-th harmonic number.) - Petros Hadjicostas, May 24 2020

			From _Petros Hadjicostas_, May 24 2020: (Start)
To illustrate the Krattenhaler-Rivoal conjecture consider the case n = 24. Then H_24 = Sum_{k=1..24} 1/k = 1347822955/356948592 and {p <= 24} = {2, 3, 5, 7, 11, 13, 17, 19, 23} with {v_p(numerator): p <= 24} = {0, 0, 1, 0, 0, 0, 0, 0, 0} and {v_p(denominator): p <= 24} = {4, 1, 0, 1, 1, 1, 1, 1, 1}.
Thus, the conjectured value for a(24) is 24! * (2^(0-4) * 3^(0-1) * 5^(1-0) * 7^(0-1) * 11^(0-1) * 13^(0-1) * 17^(0-1) * 19^(0-1) * 23^(0-1)) since no exponent of a prime is > 2. This product equals 8691005030400000 = a(24). (End)

Christian Krattenthaler, Table of n, a(n) for n = 1..40
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, 3(3) (2009), 555-591. [Part II appeared before Part I.]
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, Duke Math. J. 151(2) (2010), 175-218.

Cf. A007757, A056612, A131658.

A formula, conditional on a widely believed conjecture, can be found in Theorem 3 with k = 1 in the article by Krattenthaler and Rivoal (2007-2009) cited in the references; see the remarks before Theorem 4 in that article.

A036687 a(n) = Product_{i=0..n} (3*i+1)! / (n+i)!.

Original entry on oeis.org

1, 12, 420, 35280, 6486480, 2473511040, 1888413246720, 2815188363187200, 8043859365429888000, 43422645527382401280000, 437806134821131674998400000, 8167684917019434265210752000000, 279763767797866931083907001600000000, 17478686582471797472931336490014720000000

Offset: 0

Christian Krattenthaler (kratt(AT)ap.UniVie.AC.AT)

nonn

A049112 2-ranks of difference sets constructed from Glynn type I hyperovals.

Original entry on oeis.org

1, 1, 3, 7, 13, 23, 45, 87, 167, 321, 619, 1193, 2299, 4431, 8541, 16463, 31733, 61167, 117903, 227265, 438067, 844401, 1627635, 3137367, 6047469, 11656871, 22469341, 43311047, 83484727, 160921985, 310187099, 597904857, 1152498667

Offset: 1

Christian Krattenthaler (kratt(AT)ap.univie.ac.at)

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. Evans, H. Hollmann, C. Krattenthaler and Q. Xiang, Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, J. Combin. Theory Ser. A, 87.1 (1999), 74-119.
Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang, Supplement to "Gauss Sums, Jacobi Sums and p-ranks ..."
Q. Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.

Cf. A001595, A049114.

a:=[1,3,7,13];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]+a[n-3] +a[n-4] -1; od; Concatenation([1], a); # G. C. Greubel, Jul 10 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5) )); // G. C. Greubel, Jul 10 2019

L := 1,1,3,7,13: for i from 6 to 140 do l := nops([ L ]): L := L,op(l,[ L ])+op(l-1,[ L ])+op(l-2,[ L ])+op(l-3,[ L ])-1: od: [ L ];

Join[{1,1,3,7}, Table[a[1]=3; a[2]=1; a[3]=3; a[4]=7; a[i]=a[i-1]+a[i-2] +a[i-3]+a[i-4] -1, {i,5,40}]]
CoefficientList[Series[(1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5), {x,0,40}], x] (* G. C. Greubel, Jul 10 2019 *)

my(x='x+O('x^40)); Vec((1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5)) \\ G. C. Greubel, Jul 10 2019

((1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 10 2019

G.f.: (1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5).

a(n+1) = a(n) + a(n-1) + a(n-2) + a(n-3) - 1, n >= 5.

A049114 2-ranks of difference sets constructed from Glynn type II hyperovals.

Original entry on oeis.org

1, 1, 5, 7, 21, 37, 89, 173, 383, 777, 1665, 3441, 7277, 15159, 31885, 66645, 139865, 292757, 613823, 1285585, 2694433, 5644609, 11828501, 24782311, 51928773, 108802597, 227978105, 477674813, 1000877759, 2097121497, 4394101857

Offset: 1

Christian Krattenthaler (kratt(AT)ap.univie.ac.at)

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. Evans, H. Hollmann, C. Krattenthaler and Q. Xiang, Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, J. Combin. Theory Ser. A, 87.1 (1999), 74-119.
Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang, Supplement to "Gauss Sums, Jacobi Sums and p-ranks ..."
Q. Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.

Cf. A001595, A049112.

a:=[1,5,7,21];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2]-a[n-3] -a[n-4] +1; od; Concatenation([1], a); # G. C. Greubel, Jul 10 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5) )); // G. C. Greubel, Jul 10 2019

L := 1,1,5,7: for i from 5 to 100 do l := nops([ L ]): L := L,op(l,[ L ])+3*op(l-1,[ L ])-op(l-2,[ L ])-op(l-3,[ L ])+1: od: [ L ];

Join[{1,1,5,7}, Table[a[1]=1; a[2]=1; a[3]=5; a[4]=7; a[i]=a[i-1]+ 3*a[i-2]-a[i-3]-a[i-4] +1, {i, 5, 40}]]
CoefficientList[Series[(1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5), {x, 0, 40}], x] (* G. C. Greubel, Jul 10 2019 *)

my(x='x+O('x^40)); Vec((1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5)) \\ G. C. Greubel, Jul 10 2019

((1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 10 2019

G.f.: (1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5).

a(n+1) = a(n) + 3*a(n-1) - a(n-2) - a(n-3) + 1.

cp's OEIS Frontend

User: Christian Krattenthaler

A131658 For n >= 1, put A_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j and B_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j * (Sum__{k=j+1..j*n} (1/k)), and let u(n) be the largest integer for which exp(B_n(z)/(u(n)*A_n(z))) has integral coefficients. The sequence is u(n).

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A131657 For n >= 1, put A_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j and B_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j * (Sum_{k=1..j*n} (1/k)), and let b(n) be the largest integer for which exp(B_n(z)/(b(n)*A_n(z))) has integral coefficients. The sequence is b(n).

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A036687 a(n) = Product_{i=0..n} (3*i+1)! / (n+i)!.

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A049112 2-ranks of difference sets constructed from Glynn type I hyperovals.

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A049114 2-ranks of difference sets constructed from Glynn type II hyperovals.

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Author

Keywords

Links

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Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A131658 For n >= 1, put A_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j and B_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j * (Sum__{k=j+1..jn} (1/k)), and let u(n) be the largest integer for which exp(B_n(z)/(u(n)A_n(z))) has integral coefficients. The sequence is u(n).

A131657 For n >= 1, put A_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j and B_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j * (Sum_{k=1..jn} (1/k)), and let b(n) be the largest integer for which exp(B_n(z)/(b(n)A_n(z))) has integral coefficients. The sequence is b(n).