A102411
Even triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies !n = Sum_{i=1..k+2} T(i,k) * n^(i-1) / (2*k-2)!.
Original entry on oeis.org
0, 1, 0, -16, 5, 1, 0, 5256, -3068, 276, 32, 0, 2070720, 2367420, -912150, 53220, 3510, 0, -36031524480, 15327895296, -40587120, -387492840, 21414120, 758184, 840, -212459319878400, -75473246681280, 38182549456800, -2562251680800, -195611371200, 13639812480, 285616800, 453600
Offset: 1
Triangle starts:
0, 1, 0;
-16, 5, 1, 0;
5256, -3068, 276, 32, 0;
2070720, 2367420, -912150, 53220, 3510, 0;
-36031524480, 15327895296, -40587120, -387492840, 21414120, 758184, 840;
...
!11=4037914; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing !11.
=> !11 = [ -212459319878400 -75473246681280*11 +38182549456800*11^2 -2562251680800*11^3 -195611371200*11^4 +13639812480*11^5 +285616800*11^6 +453600*11^7 ]/10! = 4037914.
Cf.
A102412,
A094638,
A094216,
A003422,
A008276,
A101752,
A102409,
A102410,
A101751,
A000142,
A101559,
A101032,
A099731.
A102412
Odd triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+1, where k=[2*n+3+(-1)^n]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) * n^(i-1) / (2*k-2)!.
Original entry on oeis.org
0, 1, -4, 4, 0, 96, -396, 108, 0, 1012320, -192900, -64890, 11460, 90, -2038014720, 1977810240, -304486560, -12131280, 2792160, 21840, -33190735737600, 4445760574080, 2334485260800, -394554283200, 2330344800, 1198048320, 8215200
Offset: 1
Triangle starts:
0, 1;
-4, 4, 0;
96, -396, 108, 0;
1012320, -192900, -64890, 11460, 90;
-2038014720, 1977810240, -304486560, -12131280, 2792160, 21840;
...
!11=4037914; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing !11.
=> !11 = [ -33190735737600 +4445760574080*11 +2334485260800*11^2 -394554283200*11^3 +2330344800*11^4 +1198048320*11^5 +8215200*11^6 ]/10! = 4037914.
Cf.
A102411,
A094638,
A094216,
A003422,
A008276,
A101752,
A102409,
A102410,
A101751,
A000142,
A101559,
A101032,
A099731.
A102410
Odd triangle n!. This table read by rows gives the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2*n+3+(-1)^n]/4 and T(i,k) satisfies n! = Sum_{i=1..k+2} T(i,k) * n^(i-1) / (2*k-2)!.
Original entry on oeis.org
1, 0, 0, -6, 3, 1, 0, 2400, -2024, 264, 32, 0, 2570400, 909720, -666540, 55800, 3420, 0, -19071521280, 12195884736, -762499920, -282106440, 22425480, 741384, 840, -219303218534400, -11953192930560, 27128332828800, -2808016545600, -125442525600, 14164990560, 280576800
Offset: 1
Triangle starts:
1, 0, 0;
-6, 3, 1, 0;
2400, -2024, 264, 32, 0;
2570400, 909720, -666540, 55800, 3420, 0;
-19071521280, 12195884736, -762499920, -282106440, 22425480, 741384, 840;
...
11!=39916800; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing 11!.
=> 11! = [ -219303218534400 -11953192930560*11 +27128332828800*11^2 -2808016545600*11^3 -125442525600*11^4 +14164990560*11^5 +280576800*11^6 +453600*11^7 ]/10! = 39916800.
Cf.
A102409,
A008276,
A094216,
A000142,
A094638,
A101751,
A102411,
A102412,
A101752,
A003422,
A101559,
A101032,
A099731.
Showing 1-3 of 3 results.
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