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A304042
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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.
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9
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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
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OFFSET
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0,68
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LINKS
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FORMULA
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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)).
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EXAMPLE
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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;
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MATHEMATICA
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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]
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PROG
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(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);
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CROSSREFS
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Cf. A287326, A300656, A300785, A007318, A027641, A027642, A055012, A077028, A000146, A002882, A003245, A127187, A127188, A074909, A164555.
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KEYWORD
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AUTHOR
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STATUS
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approved
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