Michael David Hirschhorn - cp's OEIS frontend

User: Michael David Hirschhorn

Michael David Hirschhorn has authored 5 sequences.

A213080 Decimal expansion of Product_{n>=1} n! /(sqrt(2Pin) * (n/e)^n * (1+1/n)^(1/12)).

1, 0, 4, 6, 3, 3, 5, 0, 6, 6, 7, 7, 0, 5, 0, 3, 1, 8, 0, 9, 8, 0, 9, 5, 0, 6, 5, 6, 9, 7, 7, 7, 6, 0, 3, 7, 1, 0, 1, 9, 7, 4, 2, 1, 8, 1, 1, 3, 2, 6, 4, 4, 4, 2, 4, 4, 1, 5, 8, 7, 5, 3, 4, 0, 4, 2, 0, 3, 5, 7, 5, 1, 5, 6, 3, 7, 4, 4, 5, 7, 0, 7, 2, 5, 4, 8, 5, 8

Offset: 1

Michael David Hirschhorn, Jun 04 2012

Just as Stirling's formula for the asymptotic expansion of n! involves the constant sqrt{2 Pi}, the asymptotic expansion of the product of all binomial coefficients in a row of Pascal's triangle involves a constant, the reciprocal of the constant C defined and evaluated here.
From Bernd C. Kellner, Oct 13 2024: (Start)
It turns out that 1/C is not the complete asymptotic constant for the product of the binomial coefficients in a row of Pascal's triangle. A constant factor of (2*Pi)^(-1/4) was overlooked in the asymptotic expansion of that product given by Hirschhorn in 2013. The correct asymptotic constant is A377023.
However, the constant C equals the constant F(1) as introduced before in Kellner 2009. The constants F(1), F(2), ... occur in the same context of asymptotic constants related to asymptotic products of factorials as well as of binomial and multinomial coefficients. Moreover, the sequence (F(k)){k >= 1} is strictly decreasing with limit 1. For example, for k >= 1 the asymptotic product Prod{v >= 1} (k*v)! has the asymptotic constant F(k)*A^k*(2*Pi)^(1/4), where A = A074962 denotes the Glaisher-Kinkelin constant. Let gamma = A001620 be Euler's constant and Gamma(x) be the gamma function.
For k >= 1, the constants F(k) can be computed by an explicit formula and a divergent series expansion, as follows. We have log(F(k)) = (1/(12*k))*(1-log(k)) + (k/4)*log(2*Pi) - ((k^2+1)/k)*log(A) - Sum_{v=1..k-1} (v/k)*log(Gamma(v/k)) = gamma/(12*k) - t*zeta(3)/(360*k^3) with some t in (0,1), respectively.
It follows that log(F(1)) = 1/12 + log(2*Pi)/4 - 2*log(A) = gamma/12 - t*zeta(3)/360 with some t in (0,1), and so F(1) lies in the interval (1.0457...,1.0492...) (see Kellner 2009 and 2024). (End)

			1.04633506677050318098095065697776037101974218113264442441587534042035751563744...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, On the asymptotic behavior of Product_{k=0..n} C(n,k), Fib. Q., 51 (2013), 163-173.
Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers 9 (2009), Article #A08, 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
Bernd C. Kellner, Asymptotic products of binomial and multinomial coefficients revisited, Integers 24 (2024), Article #A59, 10 pp.; arXiv:2312.11369 [math.CO], 2023.

Cf. A074962, A000178, A084448, A241140, A272097.

Cf. A001620, A002117, A377023, A377024.

exp(2*Zeta(1,-1)-1/12)*(2*Pi)^(1/4); evalf(%,100); # Peter Luschny, Jun 22 2012

RealDigits[(Exp[1]^(1/12) (2 Pi)^(1/4))/Glaisher^2, 10, 100][[1]] (*Peter Luschny, Jun 22 2012 *)

exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4) \\ Charles R Greathouse IV, Dec 12 2013

import mpmath
mpmath.mp.pretty=True; mpmath.mp.dps = 200 #precision
mpmath.exp(2*mpmath.zeta(-1,1,1)-1/12)*(2*pi)^(1/4) # Peter Luschny, Jun 22 2012

Equals (exp(1)^(1/12)*(2*Pi)^(1/4))/A^2 where A denotes the Glaisher-Kinkelin constant.
Equals exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4).
A closely related constant is K = Product_{n>=1} (n!*(e/n)^(n+1/2))/ ((1+1/(n+1/2))^(1/12)*sqrt(2*Pi*e)) = (2^(1/6)*(3*e)^(1/12)*Pi^(1/4))/A^2 = exp(2*zeta'(-1)-1/12)*2^(1/6)*3^(1/12)*Pi^(1/4) = 1.082293504658977773529439... - Peter Luschny, Jun 22 2012
The sqrt of the constant equals Limit_{n>=1} (Product_{k=1..n-1} k!) / f(n) where f(n) = (2*Pi)^(n/2-1/8)*exp(1/24-3/4*n^2)*n^(1/2*n^2-1/12). - Peter Luschny, Jun 23 2012

A151558 Decimal expansion of the Lucas nested (A000204) radical.

Original entry on oeis.org

1, 8, 4, 0, 7, 6, 8, 3, 2, 8, 1, 4, 6, 0, 8, 2, 6, 8, 9, 8, 2, 0, 5, 7, 4, 5, 7, 7, 7, 4, 3, 5, 5, 7, 7, 8, 8, 6, 2, 1, 0, 6, 0, 3, 8, 7, 7, 7, 2, 1, 5, 4, 1, 6, 0, 5, 0, 8, 4, 9, 0, 2, 5, 0, 0, 4, 7, 8, 5, 1, 7, 4, 0, 5, 5, 6, 3, 4, 7, 0, 7, 7, 8, 5, 8, 9, 6, 0, 4, 5, 7, 0, 8, 6, 5, 1, 7, 2, 7, 8, 6, 1, 9, 5, 1

Offset: 1

Michael David Hirschhorn, May 21 2009

Analog of A105817 for the Lucas numbers (A000204).

			1.840768328146082689820574577743557788621060387772...

RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse@ LucasL@ Range@ 45], 10, 111][[1]] (* Robert G. Wilson v, May 29 2009 *)

More terms from Robert G. Wilson v, May 29 2009

A151556 Values of (n^5+47*n)/48 as n ranges over the numbers that are == +-1 mod 6.

Original entry on oeis.org

1, 70, 357, 3366, 7748, 29597, 51604, 134113, 203475, 427344, 596471, 1094240, 1444702, 2413711, 3062718, 4778067, 5884949, 8712458, 10485145, 14894314, 17595816, 24172785, 28127672, 37588181, 43189063, 56391412, 64105419, 82063428, 92438690, 116334659

Offset: 1

Michael David Hirschhorn, May 21 2009

((n^5+47n)/48)^2 is the sum of the squares of the n^2 integers from (n^4-24n^2-25)/48 to (n^4+24n^2-73)/48. For example, when n=5, 70^2 is the sum of the 25 squares of the integers from 0 to 24.

Index entries for sequences related to sums of squares

Cf. A007310, A097812, A001032.

[((2*n-1)*(81*n^4-162*n^3+144*n^2-63*n+58)+(135*n^4-270*n^3+210*n^2-75*n+26)*(-1)^n)/32: n in [1..30]]; // Vincenzo Librandi, Oct 25 2014

With[{nn=20},(#^5+47#)/48&/@Sort[Join[6Range[0,nn]+1,6Range[nn]-1]]] (* Harvey P. Dale, Nov 15 2011 *)

a(n) = a(n-1)+5*a(n-2)-5*a(n-3)-10*a(n-4)+10*a(n-5)+10*a(n-6)-10*a(n-7)-5*a(n-8)+5*a(n-9)+a(n-10)-a(n-11). [R. J. Mathar, May 21 2009]
G.f.: x*(1 +69*x +282*x^2 +2664*x^3 +2957*x^4 +7494*x^5 +2957*x^6 +2664*x^7 +282*x^8 +69*x^9 +x^10)/((1+x)^5*(x-1)^6). [R. J. Mathar, May 21 2009]
a(n) = ((2*n-1)*(81*n^4-162*n^3+144*n^2-63*n+58)+(135*n^4-270*n^3+210*n^2-75*n+26)*(-1)^n)/32. - Tani Akinari, Oct 25 2014

A143981 The number of unigraphical partitions of 2m; that is, the number of partitions of 2m which are realizable as the degree sequence of one and only one graph (where loops are not allowed but multiple edges are allowed).

Original entry on oeis.org

1, 3, 6, 9, 15, 19, 26, 36, 46, 59, 80, 100, 128, 167, 211, 267, 341, 429, 541, 682, 850, 1063, 1327, 1647, 2035, 2520, 3100, 3810, 4669, 5708, 6955, 8468, 10267, 12441, 15026, 18120, 21788, 26175, 31355, 37510, 44769, 53362, 63460, 75384, 89348

Offset: 1

Michael David Hirschhorn and James Sellers, Sep 06 2008

nonn

			For m = 4, the number of unigraphical partitions is A000041(4) + A001399(1) + A000005(5) + A083039(2) + 4 - 5 = 5 + 1 + 2 + 2 + 4 - 5 = 9.

S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. I, J. Soc. Indust. Appl. Math., vol. 10 (1962), 496-506.
S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. II. Uniqueness, J. Soc. Indust. Appl. Math., vol. 11 (1963), 135-147.

Cf. A000041, A001399, A000005, A083039.

with(combinat): with(numtheory): a:=proc(m) it:=round(m^2/12)+numbpart(m)+tau(m+1)+m-5: if m mod 6 = 0 then it:=it+2 fi: if m mod 6 = 1 then it:=it+1 fi: if m mod 6 = 2 then it:=it+3 fi: if m mod 6 = 3 then it:=it+1 fi: if m mod 6 = 4 then it:=it+2 fi: if m mod 6 = 5 then it:=it+2 fi: RETURN(it): end:

For m >= 3, a(2m) = A000041(m) + A001399(m-3) + A000005(m+1) + A083039(m-2) + m - 5.

A078757 Values of A028470(n)/A000045(n+1).

Original entry on oeis.org

1, 17, 51, 449, 1853, 12853, 61557, 382024, 1971559, 11585969, 62088471, 355111613, 1939427729, 10943439733, 60338602299, 338172377293, 1873494595072, 10464657396101, 58113694771149, 324052035315389, 1801727076022631, 10038214290617749, 55845947547948897

Offset: 1

Michael David Hirschhorn, Jan 08 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2
Index entries for linear recurrences with constant coefficients, signature (1,25,11,-47,-11,25,-1,-1).

Cf. A028470, A000045.

CoefficientList[Series[-(x^7 + x^6 - 24 x^5 + 11 x^4 + 38 x^3 - 9 x^2 - 16 x - 1) / (x^8 + x^7 - 25 x^6 + 11 x^5 + 47 x^4 - 11 x^3 - 25 x^2 - x + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 13 2014 *)
LinearRecurrence[{1,25,11,-47,-11,25,-1,-1},{1,17,51,449,1853,12853,61557,382024},30] (* Harvey P. Dale, Jul 15 2017 *)

a(n) = a(n-1)+25a(n-2)+11a(n-3)-47a(n-4)-11a(n-5)+25a(n-6)-a(n-7)-a(n-8).

G.f.: -x*(x^7+x^6-24*x^5+11*x^4+38*x^3-9*x^2-16*x-1)/(x^8+x^7-25*x^6+11*x^5+47*x^4-11*x^3-25*x^2-x+1). [Colin Barker, Jun 22 2012]

More terms from Vincenzo Librandi, Nov 13 2014

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User: Michael David Hirschhorn

A213080 Decimal expansion of Product_{n>=1} n! /(sqrt(2*Pi*n) * (n/e)^n * (1+1/n)^(1/12)).

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A151558 Decimal expansion of the Lucas nested (A000204) radical.

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A151556 Values of (n^5+47*n)/48 as n ranges over the numbers that are == +-1 mod 6.

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A143981 The number of unigraphical partitions of 2m; that is, the number of partitions of 2m which are realizable as the degree sequence of one and only one graph (where loops are not allowed but multiple edges are allowed).

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A078757 Values of A028470(n)/A000045(n+1).

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A213080 Decimal expansion of Product_{n>=1} n! /(sqrt(2Pin) * (n/e)^n * (1+1/n)^(1/12)).