André F. Labossière - cp's OEIS frontend

A119362 Combinatorial twin prime formulas. This sequence gives the coefficients a(n) of combinatorial sum formulas of n-th twin primes or lesser: twin_prime(n) = 2^(n-6)/(n-1)! Sum_{i=1..n} a(i) * C(n-1,i-1) * (i+2-n).

32, 8, 32, 52, 208, 508, 2672, 9278, 71168, 258772, 2448352, 11401798, 123001112, 660768362, 7257835148, 50721144013, 643561620832, 4610932367548, 57562797069608, 547637465534282, 7281278491404272, 71412114145523938

Offset: 1

André F. Labossière, Jul 24 2006

			twin_prime(10) = [ 2^(10-6)/(10-1)! ] * [ 32*C(10-1,0)*(-7) + 8*C(10-1,1)*(-6) + 32*C(10-1,2)*(-5) + ... + 9278*C(10-1,7)*(0) + 71168*C(10-1,8)*(1) + 258772*C(10-1,9)*(2) ] = 31

Cf. A001097, A000040, A120315, A000166, A115298, A008275.

Mathematica 5.2 - webMathematica 2 - http://library.wolfram.com/webMathematica/Education/LongDivide.jsp

A120315 Combinatorial prime formulas. This sequence gives the coefficients a(n) of combinatorial sum formulas of n-th primes or lesser: prime(n) = 2^(n-5)/(n-1)! Sum_{i=1..n} a(i) * C(n-1,i-1) * (1-(n-i)/2).

Original entry on oeis.org

32, 8, 32, 52, 208, 508, 2672, 9278, 56048, 304132, 1654552, 12649198, 79342112, 615363002, 5010269828, 43213043413, 393086195632, 3633203615548, 38586294965048, 389261740224662, 4344329090764472, 51205748753742838

Offset: 1

André F. Labossière, Jun 20 2006

			prime(7) = [ 2^(7-5)/(7-1)! ] * [ 32*C(7-1,0)*(1-(7-1)/2) + 8*C(7-1,1)*(1-(7-2)/2) + 32*C(7-1,2)*(1-(7-3)/2) + 52*C(7-1,3)*(1-(7-4)/2) + 208*C(7-1,4)*(1-(7-5)/2) + 508*C(7-1,5)*(1-(7-6)/2) + 2672*C(7-1,6)*(1-(7-7)/2) ] = 17

Cf. A000040, A007504, A000166, A115298, A008275.

A115297 Treillis triangle: a triangle read by rows showing the coefficients of sum formulas of Treillis numbers (A115298). The k-th row (k>=1) contains a(n,k) for n=1 to (k+1)/2 (odd rows) and for n=1 to k/2 (even rows), where a(n,k) satisfies Sum_{n=1..[(k+1)/2_odd, k/2_even]} a(n,k). The last term of each row (and its only odd number) equals Prime(k+1)-2.

Original entry on oeis.org

1, 3, 2, 5, 4, 9, 2, 8, 11, 6, 10, 15, 4, 8, 14, 17, 6, 12, 16, 21, 2, 10, 14, 20, 27, 6, 12, 18, 26, 29, 4, 8, 16, 24, 28, 35, 6, 12, 22, 26, 34, 39, 2, 10, 18, 24, 32, 38, 41, 6, 16, 20, 30, 36, 40, 45, 4, 12, 18, 26, 34, 38, 44, 51, 10, 14, 24, 30, 36, 42, 50, 57, 6, 12, 20, 28, 32

Offset: 1

André F. Labossière, Jan 19 2006

			The computation for obtaining the coefficients of each row of the Treillis triangle are the paired differences between primes ascending and those descending. Only half-rows are to be considered for deducing such terms.
For the 13th row:
...................19-17,.23-13,.29-11,.31-7,.37-5,.41-3,.43-2
.....................2,.....10,....18,....24,...32,...38,...41
For the 14th row:
...................19-19,.23-17,.29-13,.31-11,.37-7,.41-5,.43-3,.47-2
.....................0,.....6,.....16,....20,....30,...36,...40,...45
From _Michael Somos_, Oct 17 2016: (Start)
Triangle:
1: 1,
2: 3,
3: 2, 5,
4: 4, 9,
5: 2, 8, 11,
6: 6, 10, 15,
7: 4, 8, 14, 17,
8: 6, 12, 16, 21,
... (End)

Cf. A115298, A000040, A000166, A008276.

For odd rows:
a(1, k) = a(1, k-1) - a(1, k-2)
a(2, k) = a(1, k-1) + [ a(2, k-1) - a(2, k-2) ]
a(3, k) = a(2, k-1) + [ a(3, k-1) - a(3, k-2) ]
...
a((k-1)/2, k) = a((k-3)/2, k-1) + [ a((k-1)/2, k-1) - a((k-1)/2, k-2) ]
a((k+1)/2, k) = Prime(k) - 2
and a((k-1)/2, k-1) = Prime(k-1) - 2
a((k-1)/2, k-2) = Prime(k-2) - 2
For even rows:
a(1, k) = a(1, k-1) + [ a(2, k-1) - a(1, k-2) ]
a(2, k) = a(2, k-1) + [ a(3, k-1) - a(2, k-2) ]
a(3, k) = a(3, k-1) + [ a(4, k-1) - a(3, k-2) ]
...
a((k-2)/2, k) = a((k-2)/2, k-1) + [ a(k/2, k-1) - a((k-2)/2, k-2) ]
a(k/2, k) = Prime(k) - 2
and a(k/2, k-1) = Prime(k-1) - 2
a((k-2)/2, k-2) = Prime(k-2) - 2
The recurrent prime formulas for odd and even rows are the following : prime(k_odd) = A000040(k_odd) = A115298(k) + Sum_{n=1..(k-3)/2} [ a(n,k-2) -2*a(n,k-1) ] + A000040(k-2) - A000040(k-1) +2; prime(k_even) = A000040(k_even) = A115298(k) + Sum_{n=1..(k-2)/2} [ a(n,k-2) -a((k-2)/2,k-2) -2*a(n,k-1) +a(1,k-1) ] + A000040(k-2) - A000040(k-1) + 2

A115298 Row sums of A115297.

Original entry on oeis.org

1, 3, 7, 13, 21, 31, 43, 55, 73, 91, 115, 139, 165, 193, 227, 263, 301, 339, 381, 423, 471, 517, 569, 625, 685, 745, 809, 871, 937, 1011, 1089, 1167, 1247, 1335, 1425, 1515, 1611, 1707, 1809, 1915, 2023, 2135, 2249, 2363, 2479, 2601, 2735, 2865, 2997, 3129

Offset: 1

André F. Labossière, Jan 19 2006

Muniru A Asiru, Table of n, a(n) for n = 1..3000

Cf. A115297, A000040, A000166, A008276.

a:=n->add(ithprime(k)-ithprime(ceil(k/2)),k=2..n+1): seq(a(n),n=1..60); # Muniru A Asiru, Jan 04 2019

Accumulate[Table[Prime[n] - Prime[Ceiling[n/2]], {n, 2, 51}]] (* Jon Maiga, Jan 04 2019 *)

a(n) = Sum_{k=2..n+1} (A000040(k) - A000040(ceiling(k/2))). - Jon Maiga, Jan 04 2019

More terms from André F. Labossière, Mar 27 2006

A099731 This table shows the coefficients of sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to k, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies F(n)= Sum_{i=1..k} T(i,k) n^(k-i)/(k-1)!.

Original entry on oeis.org

1, 1, -1, 1, -5, 10, 1, -12, 59, -90, 1, -22, 203, -830, 1320, 1, -35, 525, -3985, 15374, -23640, 1, -51, 1135, -13665, 93544, -342324, 523440, 1, -70, 2170, -37870, 399889, -2542540, 8997540, -13633200, 1, -92, 3794, -90440, 1356929, -13076588, 78896236, -271996080, 409852800, 1, -117, 6198, -193410

Offset: 1

André F. Labossière, Nov 08 2004

			F(13)=233; substituting n=13 in the formula of the k-th row we obtain k=7 and the coefficients
T(i,7) will be the following: 1,-51,1135,-13665,93544,-342324,523440,
=> F(13) = [13^6-51*13^5+1135*13^4-13665*13^3+93544*13^2-342324*13+523440]/6! = 233.

Cf. A000045, A094638.

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)!.

Original entry on oeis.org

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.

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.

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.

cp's OEIS Frontend

User: André F. Labossière

A119362 Combinatorial twin prime formulas. This sequence gives the coefficients a(n) of combinatorial sum formulas of n-th twin primes or lesser: twin_prime(n) = 2^(n-6)/(n-1)! Sum_{i=1..n} a(i) * C(n-1,i-1) * (i+2-n).

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A120315 Combinatorial prime formulas. This sequence gives the coefficients a(n) of combinatorial sum formulas of n-th primes or lesser: prime(n) = 2^(n-5)/(n-1)! Sum_{i=1..n} a(i) * C(n-1,i-1) * (1-(n-i)/2).

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A115298 Row sums of A115297.

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A099731 This table shows the coefficients of sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to k, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies F(n)= Sum_{i=1..k} T(i,k) * n^(k-i)/(k-1)!.

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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 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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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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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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A099731 This table shows the coefficients of sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to k, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies F(n)= Sum_{i=1..k} T(i,k) n^(k-i)/(k-1)!.

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)!.

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).

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)!.