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 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. 9
 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 of Power Function, arXiv:1603.02468 [math.NT], 2016-2020. Petro Kolosov, Definition and table of values. Petro Kolosov, An unusual identity for odd-powers, arXiv:2101.00227 [math.GM], 2021. 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. Cf. A287326, A300656, A300785, A007318, A027641, A027642, A055012, A077028, A000146, A002882, A003245, A127187, A127188, A074909, A164555. Sequence in context: A345949 A348505 A051008 * A109768 A069293 A347398 Adjacent sequences: A304039 A304040 A304041 * A304043 A304044 A304045 KEYWORD nonn,tabl,frac AUTHOR Kolosov Petro, May 05 2018 STATUS approved

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Last modified February 5 02:00 EST 2023. Contains 360082 sequences. (Running on oeis4.)