A370416 -id:A370416 - cp's OEIS frontend

A241885 Write the coefficient of x^n/n! in the expansion of (x/(exp(x)-1))^(1/2) as f(n)/g(n); sequence gives f(n).

1, -1, 1, 1, -3, -19, 79, 275, -2339, -11813, 14217, 95265, -4634445, -193814931, 131301607, 1315505395, -3890947599, -136146236611, 46949081169401, 124889801445461, -10635113572583999, -158812278992229461, 56918172351554857, 8484151253958927197

Offset: 0

Jitender Singh, May 01 2014

For g(n) see A242225(n).
The old definition was "Numerator of (B_n)^(1/2) in the Cauchy type product (sometimes known as binomial transform) where B_n is the n-th Bernoulli number".
The Nørlund polynomials N(a, n, x) with parameter a = 1/2 evaluated at x = 0 give the rational values. - Peter Luschny, Feb 18 2024

			For n=1, B_1=-1/2 and B_1^(1/2)=-1/4 so a(1)=-1.
For n=6, B_6=1/6 and B_6^(1/2)=79/86016 so a(6)=79.
1/1, -1/4, 1/48, 1/64, -3/1280, -19/3072, 79/86016, 275/49152, -2339/2949120, -11813/1310720, 14217/11534336 = A241885 / A242225.

Robert Israel, Table of n, a(n) for n = 0..479
David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.
Jitender Singh, On an arithmetic convolution, arXiv:1402.0065 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.6.7.

Cf. A242225 (denominators), A126156, A242233.
Cf. also A222411/A222412, A350194/A350154.
Cf. A370416/A370417.

g := proc(f, n) option remember; local g0, m; g0 := sqrt(f(0));
if n=0 then g0 else if n=1 then 0 else add(binomial(n, m)*g(f,m)*g(f,n-m), m=1..n-1) fi; (f(n)-%)/(2*g0) fi end:
a := n -> numer(g(bernoulli, n));
seq(a(n), n = 0..23); # Peter Luschny, May 07 2014

a := 1
g[0] := Sqrt[f[0]]
f[k_] := BernoulliB[k]
g[1] := f[1]/(2 g[0]^1);
g[k_] := (f[k] - Sum[Binomial[k, m] g[m] g[k - m], {m, 1, k - 1}])/(2 g[0])
Table[Factor[g[k]], {k, 0, 15}] // TableForm
(* Alternative: *)
Table[Numerator@NorlundB[n, 1/2, 0], {n, 0, 23}]  (* Peter Luschny, Feb 18 2024 *)

a(n)=numerator(sum(k=0,n,binomial(-1/2,k)*binomial(n+1/2,n-k)*stirling(n+k,k,2)/binomial(n+k,k))) \\ Tani Akinari, Oct 08 2024

Theorem: a(n)/A242225(n) = n!*A222411(n)/(A222412(n)*(-1)^n/(1-2*n)) = n!*A350194(n)/(A350154(n)*(2*n+1)). - David Broadhurst, Apr 23 2022 (see Link).
For any arithmetic function f and a positive integer k > 1, define the k-th root of f to be the arithmetic function g such that g*g*...*g(k times)=f and is determined by the following recursive formula:
g(0) = f(0)^(1/m);
g(1) = f(1)/(m*g(0)^(m-1));
g(k) = 1/(m*g(0)^(m-1))*(f(k) - Sum_{k_1+...+k_m=k,k_i= 2.
This formula is applicable for any rational root of an arithmetic function with respect to the Cauchy type product.
E.g.f: sqrt(x/(exp(x)-1)); take numerators. - Peter Luschny, May 08 2014
a(n) = numerator(Sum_{k=0..n} binomial(-1/2,k)*binomial(n+1/2,n-k)*Stirling2(n+k,k)/binomial(n+k,k)). - Tani Akinari, Oct 08 2024

Simpler definition from N. J. A. Sloane, Apr 24 2022 at the suggestion of David Broadhurst

A242225 Write the coefficient of x^n/n! in the expansion of (x/(exp(x)-1))^(1/2) as f(n)/g(n); sequence gives g(n).

Original entry on oeis.org

1, 4, 48, 64, 1280, 3072, 86016, 49152, 2949120, 1310720, 11534336, 4194304, 1526726656, 2348810240, 12079595520, 3221225472, 73014444032, 51539607552, 137095356088320, 5772436045824, 3809807790243840, 725677674332160, 2023101395107840, 3166593487994880

Offset: 0

Jitender Singh, May 08 2014

For f(n) see A241885(n).
The old definition was "Denominator of (B_n)^(1/2) in the Cauchy type product (sometimes known as binomial transform) where B_n is the n-th Bernoulli number".
The Nørlund polynomials N(a, n, x) with parameter a = 1/2 evaluated at x = 0 give the rational values. - Peter Luschny, Feb 18 2024

			For n=1, B_1=-1/2 and B_1^(1/2)=-1/4 so a(1)=4.
For n=6, B_6=1/6 and B_6^(1/2)=79/86016 so a(6)=86016.

David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.
Jitender Singh, On an arithmetic convolution, arXiv:1402.0065 [math.NT], 2014.

Cf. A241885.
Cf. also A222411/A222412, A350194/A350154.
Cf. A370416/A370417.

g := proc(f, n) option remember; local g0, m; g0 := sqrt(f(0));
if n=0 then g0 else if n=1 then 0 else add(binomial(n, m)*g(f, m)*g(f, n-m), m=1..n-1) fi; (f(n)-%)/(2*g0) fi end:
a := n -> denom(g(bernoulli, n));
seq(a(n), n=0..23);

a := 1
g[0] := Sqrt[f[0]]
f[k_] := BernoulliB[k]
g[1] := f[1]/(2 g[0]^1);
g[k_] := (f[k] - Sum[Binomial[k, m] g[m] g[k - m], {m, 1, k - 1}])/(2 g[0])
Table[Denominator[Factor[g[k]]], {k, 0, 15}] // TableForm
(* Alternative: *)
Table[Denominator@NorlundB[n, 1/2, 0], {n, 0, 23}] (* Peter Luschny, Feb 18 2024 *)

Theorem: A241885(n)/A242225(n) = n!*A222411(n)/(A222412(n)*(-1)^n/(1-2*n)) = n!*A350194(n)/(A350154(n)*(2*n+1)). - David Broadhurst, Apr 23 2022 (see Link).
For any arithmetic function f and a positive integer k>1, define the k-th root of f to be the arithmetic function g such that g*g*...*g(k times)=f and is determined by the following recursive formula:
g(0)= f(0)^{1/m};
g(1)= f(1)/(mg(0)^(m-1));
g(k)= 1/(m g(0)^{m-1})*(f(k)-sum_{k_1+...+k_m=k,k_i=2.
This formula is applicable for any rational root of an arithmetic function with respect to the Cauchy type product.

Simpler definition from N. J. A. Sloane, Apr 24 2022 at the suggestion of David Broadhurst.

A370417 a(n) = denominator(N(1/2, n, 1)) where N(a, n, x) is the n-th Nørlund polynomial.

Original entry on oeis.org

1, 4, 48, 64, 1280, 1024, 86016, 16384, 2949120, 1310720, 11534336, 4194304, 1526726656, 2348810240, 2415919104, 1073741824, 73014444032, 17179869184, 27419071217664, 1924145348608, 3809807790243840, 241892558110720, 404620279021568, 351843720888320, 823314306878668800

Offset: 0

Peter Luschny, Feb 18 2024

Niels Erik Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924.

Cf. A370414, A370415, A370416.

Table[Denominator@NorlundB[n, 1/2, 1], {n, 0, 24}]

A370414 T(n, k) = numerator([x^n] N(1/2, n, x)) where N(a, n, x) is the n-th Nørlund polynomial.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, 1, 1, -3, 1, -3, 1, 1, -1, 1, -19, -3, 5, 5, -5, 1, 79, -19, -9, 5, 5, -3, 1, 275, 79, -133, -21, 35, 7, -7, 1, -2339, 275, 79, -133, -21, 7, 7, -2, 1, -11813, -2339, 825, 79, -399, -189, 21, 3, -9, 1, 14217, -11813, -2339, 1375, 395, -399, -63, 15, 15, -5, 1

Offset: 0

Peter Luschny, Feb 18 2024

Nørlund polynomials N(a, n, x) are generalizations of the powers 1, x, x^2, ... as well as of the Bernoulli polynomials 1, x - 1/2, x^2 - x + 1/6, ...
Parameter a = 0 gives the first case and a = 1 the second case. Here, we consider the case a = 1/2. You can think of it as a kind of square root of the Bernoulli polynomials. We give the coefficients of these polynomials, this sequence for the numerators, and A370415 for the denominators.
We also give the values of these polynomials at the point x = 1, which are analogous to the Bernoulli numbers; A370416 for the numerators, and A370417 for the denominators.

			The lists of rational coefficients start:
  [0] [        1]
  [1] [     -1/4,        1]
  [2] [     1/48,     -1/2,         1]
  [3] [     1/64,     1/16,      -3/4,       1]
  [4] [  -3/1280,     1/16,       1/8,      -1,     1]
  [5] [ -19/3072,   -3/256,      5/32,    5/24,  -5/4,    1]
  [6] [ 79/86016,  -19/512,    -9/256,    5/16,  5/16, -3/2,    1]
  [7] [275/49152, 79/12288, -133/1024, -21/256, 35/64, 7/16, -7/4, 1]

Niels Erik Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924.

Cf. A370415, A370416/A370417 (x=1), A241885/A242225 (x=0).

egf := (t/(exp(t) - 1))^(1/2)*exp(z*t):
ser := series(egf, t, 16): ct := n -> n!*coeff(ser, t, n):
seq(seq(numer(coeff(ct(n), z, k)), k = 0..n), n = 0..10);

Table[Numerator@CoefficientList[NorlundB[n, 1/2, x], x] , {n, 0, 10}] // Flatten

T(n, k) = numerator( n! * [z^k] [t^n] (t / (exp(t) - 1))^(1/2)*exp(z*t) ).

A370415 T(n, k) = denominator([x^n] N(1/2, n, x)) where N(a, n, x) is the n-th Nørlund polynomial.

Original entry on oeis.org

1, 4, 1, 48, 2, 1, 64, 16, 4, 1, 1280, 16, 8, 1, 1, 3072, 256, 32, 24, 4, 1, 86016, 512, 256, 16, 16, 2, 1, 49152, 12288, 1024, 256, 64, 16, 4, 1, 2949120, 6144, 3072, 384, 128, 8, 12, 1, 1, 1310720, 327680, 4096, 1024, 512, 640, 16, 4, 4, 1, 11534336, 131072, 65536, 2048, 2048, 256, 128, 8, 16, 2, 1

Offset: 0

Peter Luschny, Feb 18 2024

Niels Erik Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924.

Cf. A370414, A370416, A370417.

Table[Denominator@CoefficientList[NorlundB[n, 1/2, x], x] , {n, 0, 10}] // Flatten

cp's OEIS Frontend

A241885 Write the coefficient of x^n/n! in the expansion of (x/(exp(x)-1))^(1/2) as f(n)/g(n); sequence gives f(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A242225 Write the coefficient of x^n/n! in the expansion of (x/(exp(x)-1))^(1/2) as f(n)/g(n); sequence gives g(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A370417 a(n) = denominator(N(1/2, n, 1)) where N(a, n, x) is the n-th Nørlund polynomial.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A370414 T(n, k) = numerator([x^n] N(1/2, n, x)) where N(a, n, x) is the n-th Nørlund polynomial.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A370415 T(n, k) = denominator([x^n] N(1/2, n, x)) where N(a, n, x) is the n-th Nørlund polynomial.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica