cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A103718 Triangle of coefficients of certain polynomials used with prime numbers as variables in the computation of the array A103728.

Original entry on oeis.org

1, 2, -1, 5, -4, 1, 17, -17, 7, -1, 74, -85, 45, -11, 1, 394, -499, 310, -100, 16, -1, 2484, -3388, 2359, -910, 196, -22, 1, 18108, -26200, 19901, -8729, 2282, -350, 29, -1, 149904, -227708, 185408, -89733, 26985, -5082, 582, -37, 1, 1389456, -2199276, 1896380, -993005, 332598, -72723, 10320, -915, 46, -1
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The g.f. for the sequence {b(N,p)}, with b(N,p) the number of cyclically inequivalent two-color, N bead necklaces with p beads of one color and N-p beads of the other color is, for prime numbers p, G(p(n),x):=P(p(n)-1,x)/((1-x)^(p(n)-1)*(1-x^p(n))), with the numerator polynomial P(p(n)-1,x):= sum(r(n,k)*x^k,k=0..p(n)-1) and the row polynomials of this triangle r(n,k):=sum(a(k,m)*p(n)^m,m=0..k). p(n)=A000040(n) (prime numbers).
Row sums (signed) give A000142(k)=k!. Row sums (unsigned) coincide with A007680(k)=(2*k+1)*k!, k>=0.
The (unsigned) column sequences are, for m=0..10: A000774, A081052, A103719-A103727.

Examples

			Triangle begins:
    1;
    2,   -1;
    5,   -4,    1;
   17,  -17,    7,   -1;
   74,  -85,   45,  -11,    1;
  394, -499,  310, -100,   16,   -1;
  ...

Links

Crossrefs

Cf. A008275.

Programs

  • Mathematica
    a[0, 0] = 1; a[k_, 0] := (k - 1)! + k*a[k - 1, 0]; a[k_, m_]:= If[kIndranil Ghosh, Mar 11 2017 *)
  • PARI
    a(k, m) = if(m==0, if(k==0, 1, (k - 1)! + k*a(k - 1, 0)) , if(kIndranil Ghosh, Mar 11 2017

Formula

a(k, m) = ((-1)^m)*(|S1(k+1, m+1)| + |S1(k+1, m+2)|) = ((-1)^m)*(|S1(k+2, m+2)|-k*|S1(k+1, m+2)|), with the (signed) Stirling number triangle S1(n, m) = A048994(n, m), n >= m >= 0.
a(0, 0)=1, a(k, 0) = (k-1)! + k*a(k-1, 0); a(k, m) = -a(k-1, m-1) + k*a(k-1, m), m > 0 and a(k, m)=0 if k < m.
Let B = (n+1)-st row of Stirling cycle numbers (unsigned, A008275); say a,b,c,d,.... Then n-th row of present triangle = ((a+b), (b+c), (c+d), ..., (d)). E.g., 4th row of the Stirling cycle numbers = (6, 11, 6, 1). Then third row of A103718 = ((6+11), (11+6), (6+1), (1)) = (17, 17, 7, 1). - Gary W. Adamson, May 07 2006

Extensions

More terms from Indranil Ghosh, Mar 11 2017

A103729 Column k=2 sequence of array A103728.

Original entry on oeis.org

1, 5, 13, 41, 61, 113, 145, 221, 365, 421, 613, 761, 841, 1013, 1301, 1625, 1741, 2113, 2381, 2521, 2965, 3281, 3785, 4513, 4901, 5101, 5513, 5725, 6161, 7813, 8321, 9113, 9385, 10805, 11101, 12013, 12961, 13613, 14621, 15665, 16021
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

It is clear that the a(n) are natural numbers since only odd primes appear in the formula below.

Formula

a(n)=A103728(n+2, 2)=(1 + (p(n+2)-1)*binomial(p(n+2)-1, 2))/p(n+2), with p(n):=A000040(n) (n-th prime).
a(n)= (5 - 4*p(n+2) + p(n+2)^2)/2 = sum(A103718(k, m)*p(n+2)^m, m=0..2)/2.

A103735 Column k=8 sequence of array A103728.

Original entry on oeis.org

41, 457, 12113, 41455, 305867, 3000929, 5664121, 29442493, 75028961, 115285297, 255381053, 738339317, 1884309221, 2516676241, 5657847163, 9307388231, 11805057217, 23150085349, 35212054847, 63554702993, 131233995553
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The two a(n) formulas, given below, produce natural numbers for all n>=0.

Formula

a(n)=A103728(n+5, 8)=(1 +(p(n+5)-1)*binomial(p(n+5)-1, 8))/p(n+5), with p(n):=A000040(n) (n-th prime).
a(n)= (149904 - 227708*p(n+5) + 185408* p(n+5)^2 - 89733*p(n+5)^3 + 26985*p(n+5)^4 - 5082*p(n+5)^5 + 582*p(n+5)^6 - 37*p(n+5)^7 + 1*p(n+5)^8)/8! = sum(A103718(k, m)*p(n+5)^m, m=0..8)/8!.

A103914 Negative of column k=9 sequence of array A103728.

Original entry on oeis.org

9, 203, 10767, 46061, 475793, 6668731, 13845629, 91598867, 266769639, 435522233, 1078275557, 3609658883, 10468384561, 14540796059, 36461681717, 64117563369, 83947073543, 180056219381, 289521339853, 564930693271, 1283176845407
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The two a(n) formulas, given below, produce natural numbers for all n>=0.

Crossrefs

Cf. A103735 (k=8).

Formula

a(n)=-A103728(n+5, 9)=-(1 -(p(n+5)-1)*binomial(p(n+5)-1, 9))/p(n+5), with p(n):=A000040(n) (n-th prime).
a(n)= -(1389456 - 2199276*p(n+5) + 1896380*p(n+5)^2 - 993005*p(n+5)^3 + 332598*p(n+5)^4 - 72723*p(n+5)^5 + 10320*p(n+5)^6 - 915*p(n+5)^7 + 46*p(n+5)^8 - p(n+5)^9)/9! = -sum(A103718(k, m)*p(n+5)^m, m=0..9)/9!.

A103915 Column k=10 sequence of array A103728.

Original entry on oeis.org

1, 61, 7537, 41455, 618531, 12670589, 29075821, 247316941, 826985881, 1437223369, 3989619561, 15521533197, 51295084349, 74158059901, 207831585787, 391117136551, 528866563321, 1242387913729, 2113505780927, 4462952476841
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The two a(n) formulas, given below, produce natural numbers for all n>=0.

Crossrefs

For columns k=0..9 see A000012 (powers of 1), A040976 (primes p(n)-2), A103729-A103735, A103914.

Formula

a(n)=A103728(n+5, 10)=(1 +(p(n+5)-1)*binomial(p(n+5)-1, 10))/p(n+5), with p(n):=A000040(n) (n-th prime).
a(n)= (14257440 - 23382216*p(n+5) + 21163076*p(n+5)^2 - 11826430*p(n+5)^3 + 4318985*p(n+5)^4 - 1059828*p(n+5)^5 + 175923*p(n+5)^6 -19470*p(n+5)^7 + 1375*p(n+5)^8 - 56*p(n+5)^9 + 1*p(n+5)^10)/10! = sum(A103718(k, m)*p(n+5)^m, m=0..10)/10!.

A103730 Negative of column k=3 sequence of array A103728.

Original entry on oeis.org

3, 17, 109, 203, 527, 773, 1473, 3163, 3929, 6947, 9639, 11213, 14857, 21683, 30333, 33659, 45077, 53969, 58823, 75113, 87493, 108503, 141407, 160099, 170033, 191117, 202283, 225903, 322937, 355029, 407047, 425453, 525843, 547649, 616667, 691253
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The two a(n) formulas, given below, produce natural numbers for all n>=0.

Formula

a(n)=-A103728(n+3, 3)=-(1 -(p(n+3)-1)*binomial(p(n+3)-1, 3))/p(n+3), with p(n):=A000040(n) (n-th prime).
a(n)= -(17 - 17*p(n+3) + 7*p(n+3)^2 - p(n+3)^3)/3! = -sum(A103718(k, m)*p(n+3)^m, m=0..3)/3!.

A103731 Column k=4 sequence of array A103728.

Original entry on oeis.org

1, 13, 191, 457, 1713, 2899, 6997, 19769, 26521, 57313, 89161, 109327, 159713, 265617, 417079, 479641, 709963, 903981, 1014697, 1408369, 1727987, 2305689, 3287713, 3882401, 4208317, 4921263, 5309929, 6155857, 9930313, 11272171
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The two a(n) formulas, given below, produce natural numbers for all n>=0.

Formula

a(n)=A103728(n+3, 4)=(1 +(p(n+3)-1)*binomial(p(n+3)-1, 4))/p(n+3), with p(n):=A000040(n) (n-th prime).
a(n)= (74 - 85*p(n+3) + 45*p(n+3)^2 - 11*p(n+3)^3 + p(n+3)^4)/4! = sum(A103718(k, m)*p(n+3)^m, m=0..4)/4!.

A103732 Negative of column k=5 sequence of array A103728.

Original entry on oeis.org

5, 229, 731, 4111, 8117, 25189, 94891, 137909, 366803, 641959, 830885, 1341589, 2549923, 4504453, 5371979, 8803541, 11932549, 13799879, 20843861, 26956597, 38735575, 60493919, 74542099, 82483013, 100393765, 110446523, 132966511
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The two a(n) formulas, given below, produce natural numbers for all n>=0.

Crossrefs

Cf. A000040, A103718, A103728.

Formula

a(n) = -A103728(n+4, 5)=-(1 -(p(n+4)-1)*binomial(p(n+4)-1, 5))/p(n+4), with p(n):=A000040(n) (n-th prime).
a(n) = -(394 - 499*p(n+4) + 310*p(n+4)^2 - 100*p(n+4)^3 + 16*p(n+4)^4 - p(n+4)^5)/5! = -(Sum_{m=0..5} A103718(k, m)*p(n+4)^m)/5!.

A103733 Column k=6 sequence of array A103728.

Original entry on oeis.org

1, 191, 853, 7537, 17587, 71369, 363749, 574621, 1895149, 3744761, 5123791, 9167525, 19974397, 39789335, 49243141, 89502667, 129269281, 154098649, 253600309, 345942995, 535842121, 917491105, 1180249901, 1333475377
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The two a(n) formulas, given below, produce natural numbers for all n>=0.

Crossrefs

Cf. A000040, A103718, A103728.

Formula

a(n) = A103728(n+4, 6) = (1 +(p(n+4)-1)*binomial(p(n+4)-1, 6))/p(n+4), with p(n) := A000040(n) (n-th prime).
a(n) = (2484 - 3388*p(n+4) + 2359*p(n+4)^2 - 910*p(n+4)^3 + 196*p(n+4)^4 - 22*p(n+4)^5 + p(n+4)^6)/6! = (Sum_{m=0..6} A103718(k, m)*p(n+4)^m)/6!.

A103734 Negative of column k=7 sequence of array A103728.

Original entry on oeis.org

109, 731, 10767, 30149, 163129, 1143211, 1970129, 8122067, 18188839, 26350925, 52385857, 131260323, 295577917, 379875659, 767165717, 1181890569, 1452930119, 2608460321, 3755952517, 6277007703, 11796314207, 15849070099, 18287662313
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The two a(n) formulas, given below, produce natural numbers for all n>=0.

Formula

a(n)=-A103728(n+5, 7)=-(1 -(p(n+5)-1)*binomial(p(n+5)-1, 7))/p(n+5), with p(n):=A000040(n) (n-th prime).
a(n)= -(18108 - 26200*p(n+5) + 19901*p(n+5)^2 - 8729*p(n+5)^3 + 2282*p(n+5)^4 - 350*p(n+5)^5 + 29*p(n+5)^6 - p(n+5)^7)/7! = -sum(A103718(k, m)*p(n+5)^m, m=0..7)/7!.
Showing 1-10 of 12 results. Next

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