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 20 results. Next

A103719 Column m=2 sequence of triangle A103718(n,m), n >= 0 (without leading zeros).

Original entry on oeis.org

1, 7, 45, 310, 2359, 19901, 185408, 1896380, 21163076, 256176052, 3345574440, 46910470152, 703146058224, 11222813318544, 190069993713024, 3404916420352128, 64333503601542144, 1278725806081645056
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Crossrefs

Cf. A048994, A103718.

Formula

a(n) = |S1(n+3, 3)| + |S1(n+3, 4)| = |S1(n+4, 4)| - (n+2)*|S1(n+3, 4)|, with S1(n, m):= A048994(n, m), n >= m >= 0 (Stirling1 triangle).

A103727 Column m=10 sequence of triangle A103718(n,m), n>=0, without leading zeros.

Original entry on oeis.org

1, 67, 2795, 94822, 2893072, 83458102, 2341983852, 65055916744, 1809134090205, 50745874509291, 1443120193439925, 41755358938290270, 1232237541004281031, 37152095425581774517, 1145716456561970510800, 36166233121542624257300
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Formula

a(n)= |S1(n+11, 11)| + |S1(n+11, 12)|= |S1(n+12, 12)|-(n+10)*|S1(n+11, 12)|, with S1(n, m):= A048994(n, m), n>=>m=0 (Stirling1 triangle).

A103720 Column m=3 sequence (unsigned) of triangle A103718(n,m), n>=0, without leading zeros.

Original entry on oeis.org

1, 11, 100, 910, 8729, 89733, 993005, 11826430, 151253806, 2071221724, 30271456852, 470710866080, 7763809049424, 135443758109328, 2492613881571600, 48271966288640928, 981500863085719776
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2005

Keywords

Formula

a(n)= |S1(n+4, 4)| + |S1(n+4, 5)|= |S1(n+5, 5)|-(n+3)*|S1(n+4, 5)|, with S1(n, m):= A048994(n, m), n>=>m=0 (Stirling1 triangle).

A103721 Column m=4 sequence of triangle A103718(n,m), n>=0, without leading zeros.

Original entry on oeis.org

1, 16, 196, 2282, 26985, 332598, 4318985, 59335265, 863276986, 13293822542, 216384972440, 3716485452680, 67227576292304, 1278312555078496, 25502239872984528, 532814523875346960, 11637791340592658976
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Formula

a(n)= |S1(n+5, 5)| + |S1(n+5, 6)|= |S1(n+6, 6)|-(n+4)*|S1(n+5, 6)|, with S1(n, m):= A048994(n, m), n>=>m=0 (Stirling1 triangle).

A103722 Column m=5 sequence of triangle A103718(n,m), n>=0, without leading zeros.

Original entry on oeis.org

1, 22, 350, 5082, 72723, 1059828, 15977093, 251060381, 4127061939, 71072689688, 1282475317760, 24236090536840, 479241115418584, 9904652632613008, 213690639892631680, 4806627321727980560, 112576965096880250736
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2005

Keywords

Formula

a(n)= |S1(n+6, 6)| + |S1(n+6, 7)|= |S1(n+7, 7)|-(n+5)*|S1(n+6, 7)|, with S1(n, m):= A048994(n, m), n>=>m=0 (Stirling1 triangle).

A103723 Column m=6 sequence of triangle A103718(n,m), n>=0.

Original entry on oeis.org

1, 29, 582, 10320, 175923, 2994981, 51916865, 925979626, 17090776703, 327434340233, 6521424761488, 135100311482136, 2911046722097032, 65214540352456616, 1517981446941764000, 36684237707505024560
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Formula

a(n)= |S1(n+7, 7)| + |S1(n+7, 8)|= |S1(n+8, 8)|-(n+6)*|S1(n+7, 8)|, with S1(n, m):= A048994(n, m), n>=>m=0 (Stirling1 triangle).

A103724 Column m=7 sequence of triangle A103718(n,m), n>=0, without leading zeros.

Original entry on oeis.org

1, 37, 915, 19470, 390093, 7676097, 151706126, 3049865390, 62838757553, 1332854461081, 29179950599865, 660339422279706, 15457495745411446, 374364455260685536, 9379635007416160256, 243036207870660550192
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Formula

a(n)= |S1(n+8, 8)| + |S1(n+8, 9)|= |S1(n+9, 9)|-(n+7)*|S1(n+8, 9)|, with S1(n, m):= A048994(n, m), n>=>m=0 (Stirling1 triangle).

A103725 Column m=8 sequence of triangle A103718(n,m), n>=0, without leading zeros.

Original entry on oeis.org

1, 46, 1375, 34595, 805233, 18144126, 405723890, 9135723740, 209010337393, 4886030196762, 117128494141581, 2885780810969745, 73173111964806346, 1910999806521618802, 51421630750891773900, 1425733715141171349892
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Formula

a(n)= |S1(n+9, 9)| + |S1(n+9, 10)|= |S1(n+10, 10)|-(n+8)*|S1(n+9, 10)|, with S1(n, m):= A048994(n, m), n>=>m=0 (Stirling1 triangle).

A103726 Column m=9 sequence of triangle A103718(n,m), n>=0, without leading zeros.

Original entry on oeis.org

1, 56, 1991, 58487, 1565564, 40062022, 1006654220, 25242191260, 638127588813, 16372326795396, 428202703254105, 11449834876051845, 313619644361895091, 8810631982483310804, 254066166348007922392, 7523321707493361487300
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Formula

a(n)= |S1(n+10, 10)| + |S1(n+10, 11)|= |S1(n+11, 11)|-(n+9)*|S1(n+10, 11)|, with S1(n, m):= A048994(n, m), n>=>m=0 (Stirling1 triangle).

A103728 Coefficients of numerator polynomials of g.f.s for a certain necklace problem involving prime numbers.

Original entry on oeis.org

1, 0, 1, -1, 1, 1, -3, 5, -3, 1, 1, -5, 13, -17, 13, -5, 1, 1, -9, 41, -109, 191, -229, 191, -109, 41, -9, 1, 1, -11, 61, -203, 457, -731, 853, -731, 457, -203, 61, -11, 1, 1, -15, 113, -527, 1713, -4111, 7537, -10767, 12113, -10767, 7537, -4111, 1713, -527, 113, -15, 1, 1, -17, 145, -773, 2899, -8117, 17587
Offset: 1

Views

Author

Wolfdieter Lang, Feb 24 2005

Keywords

Comments

The row polynomials P(n,x) := Sum_{k=0..p(n)-1} a(n,k)*x^k, n >= 1, appear in the numerator of the g.f. G(p(n),x) for the numbers N(p(n),m) of inequivalent m-bead necklaces of two colors with p(n) beads of one color and m-p(n) beads of the other color. Here p(n)=A000040(n) (prime numbers). Equivalently, N(p(n),m) counts inequivalent necklaces with p(n) beads which are labeled with nonnegative numbers, such that the sum of the labels is m. For a proof of this equivalent formulation see a comment in A032191. Inequivalence is meant with respect to the cyclic group C_p(n).
This necklace g.f. is G(p(n),x) = P(n,x)/((1-x^p(n))*(1-x)^(p(n)-1)), n >= 1. The row polynomials P(n,x) are defined above. This g.f. is Z(C_p(n),x), the two variable (x[1] and x[p(n)]) cycle index polynomial for the cyclic group of prime order p(n), with substitution x[1]->1/(1-x^1)and x[p(n)]->1/(1-x^p(n)). This follows by Polya enumeration if the above mentioned labeled necklace problem is solved.
The row length sequence for this array a(n,k) is A000040(n) (n-th prime number), [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...].
The rows of this signed array are symmetric: a(n,k) = a(n,p(n)-1-k), n >= 2, k = 0..(p(n)-1)/2. See the explicit formula below.
The formulas for a(n,k), given below, produces in fact integers.
G.f. for column k, k>=0 (without leading zeros): sum(A103718(k,m)*p(n)^m,m=0..k)/k! produces for all n> pi(n) integers, where pi(n):=A000720(n), primes not exceeeding n.

Examples

			Triangle begins:
  [1, -0];
  [1, -1,  1];
  [1, -3,  5,  -3,   1];
  [1, -5, 13, -17,  13,   -5,   1];
  [1, -9, 41,-109, 191, -229, 191, -109, 41, -9, 1];
  ...
n=3: G(p(3),x)=G(5,x)=(1-3*x+5*x^2-3*x^3+1*x^4)/((1-x^5)*(1-x)^4) generates the necklace sequence A008646.
A103718(3,m), m=0..3, is [17,-17,7,-1]. Therefore (17-17*p(n)+7*p(n)^2-1*p(n)^3 )/3! gives, for n>=1, the third column [ -3,-17,-109,...].

Links

Crossrefs

The unsigned column sequences are for k=0..10: A000012 (powers of 1), A040976 (primes p(n)-2), A103729 - A103914, A103915.

Formula

a(n, k) = (1 + ((-1)^k)*(p(n)-1)*binomial(p(n)-1, k))/p(n), with p(n): = A000040(n) (n-th prime).
a(n, k) = sum(A103718(k, m)*p(n)^m, m=0..k)/k!, (row polynomials of triangle A103718 with x=p(n), divided by k!).
Showing 1-10 of 20 results. Next

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