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.

Previous Showing 11-20 of 20 results.

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!.

A158471 Stirling-like triangle by rows generated from (x-1)*(x-1)*(x-2)*(x-3)*(x-4)*...

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -7, 17, -17, 6, 1, -11, 45, -85, 74, -24, 1, -16, 100, -310, 499, -394, 120, 1, -22, 196, -910, 2359, -3388, 2484, -720, 1, -29, 350, -2282, 8729, -19901, 26200, -18108, 5040, 1, -37, 582, -5082, 26985, -89733, 185408
Offset: 0

Views

Author

Gary W. Adamson, Mar 20 2009

Keywords

Comments

Row sums of the unsigned triangle = A098558: (1, 2, 4, 12, 48, 240, 1440, 10080, ...).

Examples

			First few rows of the unsigned triangle =
1;
1, 1;
1, 2, 1;
1, 4, 5, 2;
1, 7, 17, 17, 6;
1, 11, 45, 85, 74, 24;
1, 16, 100, 310, 499, 394, 120;
1, 22, 196, 910, 2359, 3388, 2484, 720;
1, 29, 350, 2282, 8729, 19901, 26200, 18108, 5040;
1, 37, 582, 5082, 26985, 89733, 185408, 227708, 149904, 40320;
1, 46, 915, 10320, 72723, 332598, 993005, 1896380, 2199276, 1389456, 362880;
...
Example: Row 5 = x^5 - 11x^4 + 45x^3 -85x^2 + 74x - 24 =
(x-1)*(x-1)*(x-2)*(x-3)*(x-4).

Crossrefs

Cf. A098558.
Cf. A103718. - R. J. Mathar, Mar 20 2009

Formula

Triangle read by rows, n-th row = n-th degree polynomial with alternating signs generated from n terms of a*b*c*d*...; where a,b,c,... = (x-1), (x-1), (x-2), (x-3), (x-4), ... n-th row, n > 0 = charpoly of an n X n matrix with (1,1,2,3,4,...) in the diagonal and the rest zeros.

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!.
Previous Showing 11-20 of 20 results.

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