A002682
Denominators of coefficients for repeated integration.
Original entry on oeis.org
3, 45, 252, 28350, 1496880, 3405402000, 17513496000, 7815397590000, 5543722023840000, 235212205868640000, 206559082608278400000, 516914104227216696000000, 572581776990147724800000
Offset: 0
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
M:=n->(2/(2*n+1)!)*int(t*product(t^2-k^2,k=1..n),t=0..1): A:=n->((n+1)/2)*M(n)+(2*n+2)*M(n+1): seq(denom(A(n)),n=0..15); # Emeric Deutsch, Jan 25 2005
-
M[n_] := (2/(2n+1)!) Integrate[t Product[t^2-k^2, {k, 1, n}], {t, 0, 1}];
A[n_] := ((n+1)/2) M[n] + (2n+2) M[n+1];
Table[Denominator[A[n]], {n, 0, 15}] (* Jean-François Alcover, Oct 04 2021, after Maple code *)
A348220
Numerators of coefficients for numerical integration of certain differential systems (Array A(i,k) read by ascending antidiagonals).
Original entry on oeis.org
2, 2, 0, 2, 2, 1, 2, 4, 1, -1, 2, 6, 7, 0, 29, 2, 8, 19, 1, -1, -14, 2, 10, 37, 8, -1, 1, 1139, 2, 12, 61, 9, 29, 0, -37, -41, 2, 14, 91, 64, 269, -1, 1, 8, 32377, 2, 16, 127, 125, 1079, 14, 1, -1, -119, -3956, 2, 18, 169, 72, 2999, 33, -37, 0, 127, 9, 2046263
Offset: 0
Array begins:
2, 0, 1/3, -1/3, 29/90, -14/45, 1139/3780, -41/140, ...
2, 2, 1/3, 0, -1/90, 1/90, -37/3780, 8/945, ...
2, 4, 7/3, 1/3, -1/90, 0, 1/756, -1/756, ...
2, 6, 19/3, 8/3, 29/90, -1/90, 1/756, 0, ...
2, 8, 37/3, 9, 269/90, 14/45, -37/3780, 1/756, ...
2, 10, 61/3, 64/3, 1079/90, 33/10, 1139/3780, -8/945, ...
2, 12, 91/3, 125/3, 2999/90, 688/45, 13613/3780, 41/140, ...
2, 14, 127/3, 72, 6749/90, 875/18, 14281/756, 736/189, ...
2, 16, 169/3, 343/3, 13229/90, 618/5, 51031/756, 17225/756, ...
...
- Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, page 127, 1969, Arcueil. Later CELAR. Now DGA Maitrise de l'Information 35170 Bruz.
-
A[i_ /; i >= 0, k_ /; k >= 0] := A[i, k] = If[i == 0, (1/k!) Integrate[ Product[u+j, {j, -k+1, 0}], {u, -1, 1}], A[i-1, k-1] + A[i-1, k]];
A[, ] = 0;
Table[A[i-k, k] // Numerator, {i, 0, 10}, {k, 0, i}] // Flatten
-
array(nn) = {my(m = matrix(nn, nn)); for (k=0, nn-1, m[1, k+1] = bestappr(intnum(x=-1, 1, prod(j=1-k, 0, x+j)))/k!; ); for (j=1, nn-1, for (k=0, nn-1, m[j+1, k+1] = if (k>0, m[j,k], 0) + m[j, k+1];);); apply(numerator, m);} \\ Michel Marcus, Oct 08 2021
A348221
Denominators of coefficients for numerical integration of certain differential systems (Array A(i,k) read by ascending antidiagonals).
Original entry on oeis.org
1, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, 90, 1, 1, 3, 3, 90, 45, 1, 1, 3, 3, 90, 90, 3780, 1, 1, 3, 1, 90, 1, 3780, 140, 1, 1, 3, 3, 90, 90, 756, 945, 113400, 1, 1, 3, 3, 90, 45, 756, 756, 16200, 14175, 1, 1, 3, 1, 90, 10, 3780, 1, 113400, 1400, 7484400
Offset: 0
Array begins:
2, 0, 1/3, -1/3, 29/90, -14/45, 1139/3780, -41/140, ...
2, 2, 1/3, 0, -1/90, 1/90, -37/3780, 8/945, ...
2, 4, 7/3, 1/3, -1/90, 0, 1/756, -1/756, ...
2, 6, 19/3, 8/3, 29/90, -1/90, 1/756, 0, ...
2, 8, 37/3, 9, 269/90, 14/45, -37/3780, 1/756, ...
2, 10, 61/3, 64/3, 1079/90, 33/10, 1139/3780, -8/945, ...
2, 12, 91/3, 125/3, 2999/90, 688/45, 13613/3780, 41/140, ...
2, 14, 127/3, 72, 6749/90, 875/18, 14281/756, 736/189, ...
2, 16, 169/3, 343/3, 13229/90, 618/5, 51031/756, 17225/756, ...
...
- Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, page 127, 1969.
-
A[i_ /; i >= 0, k_ /; k >= 0] := A[i, k] = If[i == 0, (1/k!) Integrate[ Product[u + j, {j, -k + 1, 0}], {u, -1, 1}], A[i - 1, k - 1] + A[i - 1, k]]; A[, ] = 0;
Table[A[i - k, k] // Denominator, {i, 0, 10}, {k, 0, i}] // Flatten
Showing 1-3 of 3 results.
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