A304042 - OEIS

A304042

Triangle read by rows: T(n,k) is the denominator of R(n,k) defined implicitly by the identity Sum_{i=0..l-1} Sum_{j=0..m} R(m,j)*(l-i)^j*i^j = l^(2*m+1) holding for all l,m >= 0.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,68

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10439 (the first 144 rows of triangle)

P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.

C. Jordan, Calculus of Finite Differences, Budapest, 1939. [Annotated scans of pages 448-450 only]

Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.

Petro Kolosov, Definition and table of values.

Petro Kolosov, An unusual identity for odd-powers, arXiv:2101.00227 [math.GM], 2021.

Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.

Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Petro Kolosov, A novel proof of power rule in calculus, GitHub, 2024. See p. 5.

Petro Kolosov, Odd-power identity via multiplication of certain matrices, GitHub, 2024. See p. 4.

MathOverflow, Discussion of these coefficients, 2018.

FORMULA

Recurrence given by Max Alekseyev (see the MathOverflow link):

R(n, k) = 0 if k < 0 or k > n.

R(n, k) = (2k+1)*binomial(2k, k) if k = n.

R(n, k) = (2k+1)*binomial(2k, k)*Sum_{j=2k+1..n} R(n, j)*binomial(j, 2k+1)*(-1)^(j-1)/(j-k)*Bernoulli(2j-2k), otherwise.

T(n, k) = denominator(R(n, k)).

EXAMPLE

Triangle begins:

-----------------------------------------------------

k= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-----------------------------------------------------

n=0: 1;

n=1: 1, 1;

n=2: 1, 1, 1;

n=3: 1, 1, 1, 1;

n=4: 1, 1, 1, 1, 1;

n=5: 1, 1, 1, 1, 1, 1;

n=6: 1, 1, 1, 1, 1, 1, 1;

n=7: 1, 1, 1, 1, 1, 1, 1, 1;

n=8: 1, 1, 1, 1, 1, 1, 1, 1, 1;

n=9: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

n=10: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

n=11: 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1;

n=12: 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1;

n=13: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

n=14: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

n=15: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

MATHEMATICA

R[n_, k_] := 0

R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*

Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*

BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n

R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;

T[n_, k_] := Denominator[R[n, k]];

(* Print Fifteen Initial rows of Triangle A304042 *)

Column[ Table[ T[n, k], {n, 0, 15}, {k, 0, n}], Center]

PROG

(PARI)

up_to = 1274; \\ = binomial(50+1, 2)-1

A304042aux(n, k) = if((k<0)||(k>n), 0, (k+k+1)*binomial(2*k, k)*if(k==n, 1, sum(j=k+k+1, n, A304042aux(n, j)*binomial(j, k+k+1)*((-1)^(j-1))/(j-k)*bernfrac(2*(j-k)))));

A304042tr(n, k) = denominator(A304042aux(n, k));

A304042list(up_to) = { my(v = vector(up_to), i=0); for(n=0, oo, for(k=0, n, if(i++ > up_to, return(v)); v[i] = A304042tr(n, k))); (v); };

v304042 = A304042list(1+up_to);

A304042(n) = v304042[1+n]; \\ Antti Karttunen, Nov 07 2018

CROSSREFS

Numerators are shown in A302971.