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

Offset: 0

Author

Kolosov Petro, May 05 2018

Keywords

• nonn
• tabl
• frac

Examples

			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;
		

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, 106.37 An unusual identity for odd-powers, The Mathematical Gazette, 2022.
• 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.
• Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.
• Petro Kolosov, Discussion on coefficients of odd polynomial identity, GitHub, 2025.
• MathOverflow, Discussion of these coefficients, 2018.

Crossrefs

Numerators are shown in A302971.

Cf. A287326, A300656, A300785, A007318, A027641, A027642, A055012, A077028, A000146, A002882, A003245, A127187, A127188, A074909, A164555.

Programs

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]

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

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)).