A094216 -id:A094216 - cp's OEIS frontend

A100492 Triangle read by rows giving the coefficients of general sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies F(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

1, -1, -4, -3, 10, 49, 95, 83, 27, -90, -740, -2415, -4110, -3890, -1950, -405, 1320, 14054, 64116, 164059, 258461, 257604, 159070, 55755, 8505, -23640, -318684, -1881532, -6452300, -14294605, -21442540, -22106669, -15496012, -7078575, -1905120, -229635, 523440, 8474100, 61424596

Offset: 1

André F. Labossière, Nov 22 2004

			F(7) = (1/(7-1)!) * [ 7^(7-1) -{1+4*(7-2)+3*C(7-2,2)}*7^(7-2) +{10+49*(7-3)+95*C(7-3,2)+83*C(7-3,3) +27*C(7-3,4)}*7^(7-3) -{90+740*(7-4)+2415*C(7-4,2)+4110*C(7-4,3)}*7^(7-4) +... ]
= (1/6!) * [ 7^6 -{1+20+30}*7^5 +{10+196+570+332+27}*7^4 -{90+2220+7245+4110}*7^3 +{1320+28108 +64116}*7^2 -{23640+318684}*7 +{523440} ]
= (1/6!) * [ 7^6 -51*7^5 +1135*7^4 -13665*7^3 +93544*7^2 -342324*7 +523440 ]
= (1/720) * [ 117649 -857157 +2725135 -4687095 +4583656 -2396268 +523440 ] = 9360/720 = 13.

Cf. A099731, A000045, A094216, A094638, A000108.

A101559 This table (read by rows) shows the coefficients of sum formulas of n-th subfactorial numbers (A000166). The n-th row (n>=1) contains T(i,n) for i=1 to n, where T(i,n) satisfies Subf(n) = Sum_{i=1..n} T(i,n) * n^(n-i).

Original entry on oeis.org

1, 1, -2, 1, -4, 4, 1, -7, 15, -10, 1, -11, 42, -65, 34, 1, -16, 96, -267, 339, -154, 1, -22, 191, -831, 1891, -2103, 874, 1, -29, 344, -2151, 7600, -15023, 15171, -5914, 1, -37, 575, -4880, 24600, -74884, 133147, -124755, 46234, 1, -46, 907, -10025, 68153, -293925, 798564, -1305847, 1151331, -409114, 1, -56

Offset: 1

André F. Labossière, Dec 06 2004

			Subf(9) = [ 9^8 -37*9^7 +575*9^6 -4880*9^5 +24600*9^4 -74884*9^3 +133147*9^2 - 124755*9 +46234 ] = 14833.

Cf. A101560, A000166, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.

A101752 Table (read by rows) giving the coefficients of sum formulas of n-th Left factorials (A003422).

Original entry on oeis.org

1, 0, 1, 5, -16, 8, 69, -767, 1314, 117, 1774, -30405, 78914, 69024

Offset: 1

André F. Labossière, Dec 17 2004

The k-th row (k>=1) contains T(i,k) for i=1 to k+1, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) * n^(k-i+1) / k!.

			!7 = 874; substituting n=7 in the formula of the k-th row we obtain k=4 and the coefficients T(i,4) will be the following: 117,1774,-30405,78914,69024, => !7 = [ 117*7^4 +1774*7^3 -30405*7^2 +78914*7 +69024 ]/4! = 874.

Cf. A094216.

A101033 Triangle read by rows giving the coefficients of general sum formulas of n-th Lucas numbers (A000204). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies L(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

Original entry on oeis.org

1, 1, -2, -3, 2, 15, 51, 65, 27, 6, -148, -945, -2292, -2776, -1680, -405, 24, 2290, 19580, 71965, 145525, 175244, 125950, 50085, 8505, 120, -41676, -473072, -2340400, -6676835, -12132890, -14587261, -11619692, -5290005, -1752030, -229635, 720, 943908, 13132532, 81977672, 303352938, 740797855

Offset: 1

André F. Labossière, Nov 30 2004

			L(7)= (1/(7-1)!) * [ 7^(7-1) -{-1+2*(7-2)+3*C(7-2,2)}*7^(7-2) +{2+15*(7-3)+51*C(7-3,2)+65*C(7-3,3) +27*C(7-3,4)}*7^(7-3) -{-6+148*(7-4)+945*C(7-4,2)+2292*C(7-4,3)}*7^(7-4) +... ]
= (1/6!) * [ 7^6 -{-1+10+30}*7^5 +{2+60+306+260+27}*7^4 -{-6+444+2835+2292}*7^3 +{24+4580+19580}*7^2 -{-120+41676}*7 +{720} ] = (1/6!) * [ 7^6 -39*7^5 +655*7^4 -5565*7^3 +24184*7^2 -41556*7 +720 ]
= (1/720) * [ 117649 -655473 +1572655 -1908795 +1185016 -290892 +720 ] = 20880/720 = 29.

Ron Knott, The Lucas Numbers in Pascal's Triangle.

Cf. A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.

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=[2n+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

André F. Labossière, Jan 07 2005

The sum of signed coefficients for each k-th row is divisible by (2*k-2)!. Moreover, another variant (but an incomplete one, and sorted differently) of the above sequence is presented in A101752.

			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=[2n+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

André F. Labossière, Jan 07 2005

Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!.

			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.

A101751 Table (read by rows) giving the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (k>=1, n>=2) contains T(i,k) for i=1 to k+1, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies Fact(n) = Sum_{i=1..k+1} T(i,k) (n-1)^(k-i+1) / (2*k-2)!.

Original entry on oeis.org

1, 0, 1, 3, -6, 32, 264, -2024, 2400, 3420, 55800, -666540, 909720, 2570400, 90440, 13101144, 72406040, -3757930680, 13117344800, 72965762016, -261763004160

Offset: 1

André F. Labossière, Dec 17 2004

			Fact(8) = 5040; substituting n=8 in the formula of the k-th row we obtain k=4 and the coefficients
T(i,4) will be the following: 3420,55800,-666540,909720,2570400, => Fact(8) = [ 3420*7^4 +55800*7^3 -666540*7^2 +909720*7 +2570400 ]/6! = 7! =5040.

Cf. A008276, A094216, A000142, A094638, A101752, A003422, A101559, A101032, A099731.

A102409 Even 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+3, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies n! = Sum_{i=1..k+3} T(i,k) n^(i-1) / (2*k-2)!.

Original entry on oeis.org

0, 1, 0, 0, 0, -20, 8, 0, 0, 20280, -6530, -1275, 362, 3, 0, -8749440, 21627600, -4871940, -66510, 48300, 390, 0, -261763004160, 72965762016, 13117344800, -3757930680, 72406040, 13101144, 90440, 0, -974260634054400, -1140185248443360, 353509119454680, -8136128999880, -3234018579750

Offset: 1

André F. Labossière, Jan 07 2005

Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!. Moreover, another variant (but an incomplete one, and sorted differently) of the above sequence is presented in A101751.

			Triangle starts:
0, 1, 0, 0;
0, -20, 8, 0, 0;
20280, -6530, -1275, 362, 3, 0;
-8749440, 21627600, -4871940, -66510, 48300, 390, 0;
-261763004160, 72965762016, 13117344800, -3757930680, 72406040, 13101144, 90440, 0;
...
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! = [ -974260634054400 -1140185248443360*11 +353509119454680*11^2 -8136128999880*11^3 -3234018579750*11^4 +109743298560*11^5 +6053880420*11^6 +34067880*11^7 +9450*11^8 ]/10! = 39916800.

Cf. A102410, A008276, A094216, A000142, A094638, A101751, A102411, A102412, A101752, A003422, 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=[2n+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

André F. Labossière, Jan 07 2005

Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!.

			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.

A101560 Triangle read by rows giving the coefficients of general sum formulas of n-th Subfactorial numbers (A000166). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies Subf(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k).

Original entry on oeis.org

1, -2, -2, -1, 4, 11, 16, 11, 3, -10, -55, -147, -215, -179, -80, -15, 34, 305, 1247, 2910, 4224, 3904, 2245, 735, 105, -154, -1949, -10971, -35970, -76269, -109554, -108184, -72639, -31780, -8190, -945, 874, 14297, 103679, 443762, 1255671, 2484619, 3535727, 3654132, 2726787, 1434797

Offset: 1

André F. Labossière, Dec 06 2004

			Subf(7) = 7^(7 - 1) - {2 + 2*(7 - 2) + C(7 - 2,2)}*7^(7 - 2) + {4 + 11*(7 - 3) + 16*C(7 - 3,2) + 11*C(7 - 3,3) + 3*C(7 - 3,4)}*7^(7 - 3) - {10 + 55*(7 - 4) + 147*C(7 - 4,2) + 215*C(7 - 4,3)}*7^(7 - 4) + ...
= 7^6 - {2 + 10 + 10}*7^5 + {4 + 44 + 96 + 44 + 3}*7^4 - {10 + 165 + 441 + 215}*7^3 + {34 + 610 + 1247}*7^2 - {154 + 1949}*7 + {874}
= 7^6 - 22*7^5 + 191*7^4 - 831*7^3 + 1891*7^2 - 2103*7 + 874
= 117649 - 369754 + 458591 - 285033 + 92659 - 14721 + 874 = 265.

Cf. A101559, A000166, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.

cp's OEIS Frontend

A100492 Triangle read by rows giving the coefficients of general sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k-1, where T(i,k) satisfies F(n) = Sum_{k=1..n} Sum_{i=1..2*k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

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A101559 This table (read by rows) shows the coefficients of sum formulas of n-th subfactorial numbers (A000166). The n-th row (n>=1) contains T(i,n) for i=1 to n, where T(i,n) satisfies Subf(n) = Sum_{i=1..n} T(i,n) * n^(n-i).

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A101752 Table (read by rows) giving the coefficients of sum formulas of n-th Left factorials (A003422).

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A101033 Triangle read by rows giving the coefficients of general sum formulas of n-th Lucas numbers (A000204). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k-1, where T(i,k) satisfies L(n) = Sum_{k=1..n} Sum_{i=1..2*k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

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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)!.

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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)!.

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A101751 Table (read by rows) giving the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (k>=1, n>=2) contains T(i,k) for i=1 to k+1, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies Fact(n) = Sum_{i=1..k+1} T(i,k) * (n-1)^(k-i+1) / (2*k-2)!.

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A102409 Even 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+3, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies n! = Sum_{i=1..k+3} T(i,k) * n^(i-1) / (2*k-2)!.

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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)!.

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A101560 Triangle read by rows giving the coefficients of general sum formulas of n-th Subfactorial numbers (A000166). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k-1, where T(i,k) satisfies Subf(n) = Sum_{k=1..n} Sum_{i=1..2*k-1} T(i,k) * C(n-k,i-1) * n^(n-k).

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A100492 Triangle read by rows giving the coefficients of general sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies F(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

A101033 Triangle read by rows giving the coefficients of general sum formulas of n-th Lucas numbers (A000204). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies L(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

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=[2n+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)!.

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=[2n+3+(-1)^n]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) n^(i-1) / (2*k-2)!.

A101751 Table (read by rows) giving the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (k>=1, n>=2) contains T(i,k) for i=1 to k+1, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies Fact(n) = Sum_{i=1..k+1} T(i,k) (n-1)^(k-i+1) / (2*k-2)!.

A102409 Even 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+3, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies n! = Sum_{i=1..k+3} T(i,k) n^(i-1) / (2*k-2)!.

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=[2n+3+(-1)^n]/4 and T(i,k) satisfies n! = Sum_{i=1..k+2} T(i,k) n^(i-1) / (2*k-2)!.

A101560 Triangle read by rows giving the coefficients of general sum formulas of n-th Subfactorial numbers (A000166). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies Subf(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k).