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.

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A056001 a(n) = (n+1)*binomial(n+7, 7).

Original entry on oeis.org

1, 16, 108, 480, 1650, 4752, 12012, 27456, 57915, 114400, 213928, 381888, 655044, 1085280, 1744200, 2728704, 4167669, 6229872, 9133300, 13156000, 18648630, 26048880, 35897940, 48859200, 65739375, 87512256, 115345296, 150629248, 195011080, 250430400, 319159632
Offset: 0

Views

Author

Barry E. Williams, Jun 18 2000

Keywords

Comments

Original name: A second-order recursive sequence.

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Partial sums of A052226.
Cf. A093565 ((8, 1) Pascal, column m=8).
Cf. A007318, A000142, A245334.

Programs

Formula

G.f.: (1+7*x)/(1-x)^9.
a(n) = A245334(n+7,7)/A000142(7). - Reinhard Zumkeller, Aug 31 2014
a(n) = A000581(n+8)+7*A000581(n+7). - R. J. Mathar, Oct 24 2014
E.g.f.: (5040 +75600*x +194040*x^2 +170520*x^3 +66150*x^4 +12642*x^5 + 1225*x^6 +57*x^7 +x^8)*exp(x)/5040. - G. C. Greubel, Aug 29 2019
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 7*Pi^2/6 - 37583/3600.
Sum_{n>=0} (-1)^n/a(n) = 7*Pi^2/12 - 2912*log(2)/15 + 155701/1200. (End)

A050404 Partial sums of A051878.

Original entry on oeis.org

1, 14, 77, 280, 798, 1932, 4158, 8184, 15015, 26026, 43043, 68432, 105196, 157080, 228684, 325584, 454461, 623238, 841225, 1119272, 1469930, 1907620, 2448810, 3112200, 3918915, 4892706, 6060159, 7450912
Offset: 0

Views

Author

Barry E. Williams, Dec 21 1999

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A051878.
Cf. A093565 ((8, 1) Pascal, column m=6).

Programs

Formula

a(n) = binomial(n+5, 5)*(4*n+3)/3.
G.f.: (1+7*x)/(1-x)^7.
E.g.f.: (360 +4680*x +9000*x^2 +5400*x^3 +1275*x^4 +123*x^5 +4*x^6 )*exp(x)/360. - G. C. Greubel, Aug 30 2019

Extensions

Corrected by T. D. Noe, Nov 09 2006

A052226 Partial sums of A050404.

Original entry on oeis.org

1, 15, 92, 372, 1170, 3102, 7260, 15444, 30459, 56485, 99528, 167960, 273156, 430236, 658920, 984504, 1438965, 2062203, 2903428, 4022700, 5492630, 7400250, 9849060, 12961260, 16880175, 21772881, 27833040, 35283952, 44381832, 55419320, 68729232, 84688560, 103722729, 126310119
Offset: 0

Views

Author

Barry E. Williams, Jan 29 2000

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
  • Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Links

Crossrefs

Cf. A050404.
Cf. A093565 ((8, 1) Pascal, column m=7).

Programs

  • GAP
    List([0..40], n-> (8*n+7)*Binomial(n+6, 6)/7); # G. C. Greubel, Aug 29 2019
  • Magma
    [(8*n+7)*Binomial(n+6, 6)/7: n in [0..40]]; // G. C. Greubel, Aug 29 2019
  • Maple
    seq((8*n+7)*Binomial(n+6, 6)/7, n=0..40); # G. C. Greubel, Aug 29 2019
  • Mathematica
    Table[(8*n+7)*Binomial[n+6, 6]/7, {n,0,40}] (* G. C. Greubel, Aug 29 2019 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,15,92,372,1170,3102,7260,15444},40] (* Harvey P. Dale, Aug 12 2021 *)
  • PARI
    vector(40, n, (8*n-1)*binomial(n+5, 6)/7) \\ G. C. Greubel, Aug 29 2019
  • Sage
    [(8*n+7)*binomial(n+6, 6)/7 for n in (0..40)] # G. C. Greubel, Aug 29 2019

Formula

a(n) = (8*n+7)*C(n+6, 6)/7.
G.f.: (1+7*x)/(1-x)^8.
E.g.f.: (5040 +70560*x +158760*x^2 +117600*x^3 +36750*x^4 +5292*x^5 +343*x^6 +8*x^7)*exp(x)/5040. - G. C. Greubel, Aug 29 2019

Extensions

Terms a(25) onward added by G. C. Greubel, Aug 29 2019

A056122 a(n) = (8*n+9)*C(n+8,8)/9.

Original entry on oeis.org

1, 17, 125, 605, 2255, 7007, 19019, 46475, 104390, 218790, 432718, 814606, 1469650, 2554930, 4299130, 7027834, 11195503, 17425375, 26558675, 39714675, 58363305, 84412185, 120310125, 169169325, 234908700, 322420956, 437766252
Offset: 0

Views

Author

Barry E. Williams, Jul 06 2000

Keywords

Links

Crossrefs

Cf. A056001.
Cf. A093565 ((8, 1) Pascal, column m=9). Partial sums of A056001.
Cf. similar sequences listed in A254142.

Programs

  • GAP
    List([0..40], n-> (8*n+9)*Binomial(n+8,8)/9); # G. C. Greubel, Aug 29 2019
  • Magma
    [(8*n+9)*Binomial(n+8,8)/9: n in [0..40]]; // G. C. Greubel, Aug 29 2019
  • Maple
    seq((8*n+9)*binomial(n+8,8)/9, n=0..40); # G. C. Greubel, Aug 29 2019
  • Mathematica
    Table[(8n+9) Binomial[n+8,8]/9,{n,0,40}]  (* Harvey P. Dale, Mar 09 2011 *)
  • PARI
    vector(40, n, (8*n+1)*binomial(n+7,8)/9) \\ G. C. Greubel, Aug 29 2019
  • Sage
    [(8*n+9)*binomial(n+8,8)/9 for n in (0..40)] # G. C. Greubel, Aug 29 2019

Formula

G.f.: (1+7*x)/(1-x)^10.
a(n) = (362880 + 1308816*n + 1939788*n^2 + 1550548*n^3 + 740313*n^4 + 220416*n^5 + 41202*n^6 + 4692*n^7 + 297*n^8 + 8*n^9)/362880. - Harvey P. Dale, Mar 09 2011
E.g.f.: (362880 +5806080*x +16692480*x^2 +16934400*x^3 +7832160*x^4 + 1862784*x^5 +239904 x^6 +16704*x^7 +585*x^8 +8*x^9)*exp(x)/362880. - G. C. Greubel, Aug 29 2019
Previous Showing 11-14 of 14 results.

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