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 1-3 of 3 results.

A247613 a(n) = Sum_{k=0..8} binomial(16,k)*binomial(n,k).

Original entry on oeis.org

1, 17, 153, 969, 4845, 20349, 74613, 245157, 735471, 2031535, 5189327, 12316239, 27322191, 57029103, 112740255, 212383935, 383358645, 666220005, 1119362365, 1824861005, 2895653673, 4484253081, 6793194849, 10087438257, 14708950035, 21093714291
Offset: 0

Views

Author

Vincenzo Librandi, Sep 23 2014

Keywords

Links

Crossrefs

Cf. A005408, A056108, A247608 - A247612.

Programs

  • Magma
    m:=8; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]];
  • Magma
    [(20160-15076944*n+40499716*n^2-42247940*n^3 +23174515*n^4-7234136*n^5+1335334*n^6-134420*n^7 +6435*n^8)/20160: n in [0..40]];
  • Mathematica
    Table[(20160 - 15076944 n + 40499716 n^2 - 42247940 n^3 + 23174515 n^4 - 7234136 n^5 + 1335334 n^6 - 134420 n^7 + 6435 n^8)/20160, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 8 x + 36 x^2 + 120 x^3 + 330 x^4 + 792 x^5 + 1716 x^6 + 3432 x^7 + 6435 x^8)/(1 - x)^9, {x, 0, 40}], x]
Table[Sum[Binomial[16,k]Binomial[n,k],{k,0,8}],{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,17,153,969,4845,20349,74613,245157,735471},40] (* Harvey P. Dale, Mar 25 2015 *)
  • Sage
    m=8; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 23 2014

Formula

G.f.: (1 + 8*x + 36*x^2 + 120*x^3 + 330*x^4 + 792*x^5 + 1716*x^6 + 3432*x^7 + 6435*x^8) / (1-x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
a(n) = (20160 - 15076944*n + 40499716*n^2 - 42247940*n^3 + 23174515*n^4 - 7234136*n^5 + 1335334*n^6 - 134420*n^7 + 6435*n^8) / 20160.

A247614 a(n) = Sum_{k=0..9} binomial(18,k)*binomial(n,k).

Original entry on oeis.org

1, 19, 190, 1330, 7315, 33649, 134596, 480700, 1562275, 4686825, 13079352, 34084128, 83204745, 191006115, 414237570, 852920310, 1675575165, 3155247975, 5719519850, 10018268150, 17013571223, 28096825757, 45238870040, 71179679480, 109665022415
Offset: 0

Views

Author

Vincenzo Librandi, Sep 23 2014

Keywords

Links

Crossrefs

Cf. A005408, A056108, A247608 - A247612.

Programs

  • Magma
    m:=9; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]];
  • Magma
    [(181440+462101904*n-1283316876*n^2+1433031524*n^3 -853620201*n^4+303063726*n^5-66245634*n^6 +8905416*n^7-678249*n^8+24310*n^9)/181440: n in [0..40]];
  • Mathematica
    Table[(181440 + 462101904 n - 1283316876 n^2 + 1433031524 n^3 - 853620201 n^4 + 303063726 n^5 - 66245634 n^6 + 8905416 n^7 - 678249 n^8 + 24310 n^9)/181440, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 9 x + 45 x^2 + 165 x^3 + 495 x^4 + 1287 x^5 + 3003 x^6 + 6435 x^7 + 12870 x^8 + 24310 x^9)/(1 - x)^10, {x, 0, 40}], x]
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,19,190,1330,7315,33649,134596,480700,1562275,4686825},30] (* Harvey P. Dale, Jul 19 2019 *)
  • Sage
    m=9; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 23 2014

Formula

G.f.: (1 +9*x +45*x^2 +165*x^3 +495*x^4 +1287*x^5 +3003*x^6 + 6435*x^7 +12870*x^8 +24310*x^9) / (1-x)^10.
a(n) = 10*a(n-1) -45*a(n-2) +120*a(n-3) -210*a(n-4) +252*a(n-5) -210*a(n-6) +120*a(n-7) -45*a(n-8) +10*a(n-9) -a(n-10).
a(n) = (181440 + 462101904*n - 1283316876*n^2 + 1433031524*n^3 - 853620201*n^4 + 303063726*n^5 - 66245634*n^6+8905416*n^7 - 678249*n^8 + 24310*n^9) / 181440.

A247615 a(n) = Sum_{k=0..10} binomial(20,k)*binomial(n,k).

Original entry on oeis.org

1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015, 84504355, 223651350, 558350430, 1318250890, 2952624906, 6296642121, 12834146941, 25098124271, 47262174531, 85990654178, 151631858378, 259857912678, 433877085278, 707369215553
Offset: 0

Views

Author

Vincenzo Librandi, Sep 23 2014

Keywords

Links

Crossrefs

Cf. A005408, A056108, A247608, A247609, A247610, A247611, A247612.

Programs

  • Magma
    m:=10; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]];
  • Magma
    I:=[1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015]; [n le 11 select I[n] else 11*Self(n-1) -55*Self(n-2) +165*Self(n-3) -330*Self(n-4) +462*Self(n-5) -462*Self(n-6) +330*Self(n-7) -165*Self(n-8) +55*Self(n-9) -11*Self(n-10) +Self(n-11): n in [1..40]];
  • Mathematica
    CoefficientList[Series[(1 + 10 x + 55 x^2 + 220 x^3 + 715 x^4 + 2002 x^5 + 5005 x^6 + 11440 x^7 + 24310 x^8 + 48620 x^9 + 92378 x^10)/(1 - x)^11, {x, 0, 40}], x]
Table[Sum[Binomial[20,k]Binomial[n,k],{k,0,10}],{n,0,30}] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,21,231,1771,10626,53130,230230,888030,3108105,10015005,30045015},30] (* Harvey P. Dale, May 19 2015 *)
  • Sage
    m=10; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 23 2014

Formula

G.f.: (1 + 10*x + 55*x^2 + 220*x^3 + 715*x^4 + 2002*x^5 + 5005*x^6 + 11440*x^7 + 24310*x^8 + 48620*x^9 + 92378*x^10) / (1-x)^11.
a(n) = 11*a(n-1) -55*a(n-2) +165*a(n-3) -330*a(n-4) +462*a(n-5) -462*a(n-6)+330*a(n-7) -165*a(n-8) +55*a(n-9) -11*a(n-10) +a(n-11).
a(n) = 1 - 5512999*n/630 + 212329883*n^2/8400 - 134689309*n^3/4536 + 3453077689*n^4/181440 - 64212077*n^5/8640 + 80300707*n^6/43200 - 1817521*n^7/6048 + 1860157*n^8/60480 - 331721*n^9/181440 + 46189*n^10/907200.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.