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 1-6 of 6 results.

A001801 Coefficients of Legendre polynomials.

Original entry on oeis.org

3, 15, 105, 315, 6930, 18018, 90090, 218790, 2078505, 4849845, 22309287, 50702925, 1825305300, 4071834900, 18032411700, 39671305740, 347123925225, 755505013725, 3273855059475, 7064634602025, 121511715154830, 260382246760350, 1112542327066950, 2370198870707850, 20146690401016725
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
  • G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001790, A001796, A008316, A011371.
Bisection of A004733.
Diagonal 3 of triangle A100258.

Programs

  • Magma
    A001801:= func< n | 3*Binomial(n+3,3)*Catalan(n+2)*2^(Valuation(Factorial(n+4),2)-n-4) >;
[A001801(n): n in [0..30]]; // G. C. Greubel, Apr 26 2025
  • Mathematica
    A001801[n_]:= 3*2^(2*n+1)*Binomial[n+3/2, n]/2^DigitCount[n+4,2,1];
Table[A001801[n], {n,0,40}] (* G. C. Greubel, Apr 26 2025 *)
  • PARI
    a(n)=if(n<0,0,polcoeff(pollegendre(n+4),n)*2^valuation((n\2*2+4)!,2))
  • SageMath
    def A001801(n): return 3*2^(n-3)*binomial(n+3/2,n)*2^valuation(factorial(n+4), 2)
print([A001801(n) for n in range(31)]) # G. C. Greubel, Apr 26 2025

Formula

a(n) = 3*2^(n-3)*binomial(n + 3/2, n)*2^A011371(n+4). - G. C. Greubel, Apr 26 2025

Extensions

More terms from Michael Somos, Oct 25 2002

A001800 Coefficients of Legendre polynomials.

Original entry on oeis.org

1, 3, 30, 70, 315, 693, 12012, 25740, 109395, 230945, 1939938, 4056234, 16900975, 35102025, 1163381400, 2404321560, 9917826435, 20419054425, 167890003050, 344616322050, 1412926920405, 2893136075115, 47342226683700, 96742811049300, 395033145117975
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
  • G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001790, A001796, A001803, A008316.
Diagonal 2 of triangle A100258.

Programs

  • Magma
    A001800:= func< n | (n+1)*(n+2)*Catalan(n+1)/2^(&+Intseq(n+2, 2)) >;
[A001800(n): n in [0..30]]; // G. C. Greubel, Apr 25 2025
  • Maple
    wt:= proc(n) local m, r; m:=n; r:=0;
       while m>0 do r:= r+irem(m, 2, 'm') od; r
     end:
a:= n-> (n+1) *binomial(2*n+2, n+1)/2^wt(n+2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 29 2013
  • Mathematica
    a[n_] := (n+1)*Binomial[2*n+2, n+1]/2^DigitCount[n+2, 2, 1]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 13 2014 *)
  • PARI
    a(n)=if(n<0,0,-polcoeff(pollegendre(n+2),n)*2^valuation((n\2*2)!,2))
  • SageMath
    def A001800(n): return (n+1)*binomial(2*n+2,n+1)//2^sum((n+2).digits(2))
print([A001800(n) for n in range(31)]) # G. C. Greubel, Apr 25 2025

Formula

a(n) = (n+1) * C(2n+2, n+1) / 2^A000120(n+2).

Extensions

More terms from Michael Somos, Oct 25 2002

A001802 Coefficients of Legendre polynomials.

Original entry on oeis.org

5, 35, 1260, 4620, 30030, 90090, 1021020, 2771340, 14549535, 37182145, 1487285800, 3650610600, 17644617900, 42075627300, 396713057400, 925663800600, 4281195077775, 9821565178425, 178970743251300, 405039050516100, 1822675727322450, 4079321865912150
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
  • G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Diagonal 4 of triangle A100258.
Cf. A001796, A011371.

Programs

  • Magma
    A001802:= func< n | Binomial(n+4,4)*Catalan(n+3)*2^(Valuation(Factorial(n+6),2)-n-4) >;
[A001802(n): n in [0..30]]; // G. C. Greubel, Apr 26 2025
  • Mathematica
    A001802[n_]:= 5*4^(n+1)*Binomial[n+5/2,n]/2^DigitCount[n+6,2,1];
Table[A001802[n], {n,0,30}] (* G. C. Greubel, Apr 26 2025 *)
  • PARI
    a(n)= - polcoeff(pollegendre(n+6), n)*2^valuation((n\2*2+6)!, 2) \\ Michel Marcus, May 29 2013
  • SageMath
    def A001802(n): return 5*2^(n-4)*binomial(n+5/2,n)*2^valuation(factorial(n+6), 2)
print([A001802(n) for n in range(31)]) # G. C. Greubel, Apr 26 2025

Formula

a(n) = 5*2^(n-4)*binomial(n+5/2, n)*2^A011371(n+6). - G. C. Greubel, Apr 26 2025

Extensions

More terms from Michel Marcus, Feb 02 2015

A001797 Coefficients of Legendre polynomials.

Original entry on oeis.org

2, 20, 110, 2600, 16150, 208012, 1376550, 74437200, 511755750, 7134913500, 50315410002, 1433226830360, 10292051290430, 148889972762300, 1083802983548950, 126935005433253024, 933787075442258310, 13799767368300523260
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficient of Legendre_2(x) when x^n is written in term of Legendre polynomials. - Sean A. Irvine, Nov 28 2012

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001796.

Programs

  • Magma
    B:= Binomial;
A001797:= func< n | 20*B(n+1,2)*Numerator(B(4*n,2*n)/2^(4*n))/(3*B(2*n+3,3)) >;
[A001797(n): n in [1..30]]; // G. C. Greubel, Apr 23 2025
  • Maple
    a:=n->(10*n/((2*n+1)*(2*n+3)))*numer(binomial(4*n,2*n)/2^(4*n)); # Sean A. Irvine, Nov 28 2012
  • Mathematica
    A001797[n_]:= With[{B=Binomial}, 20*B[n+1,2]*Numerator[B[4*n,2*n]/2^(4*n)]/( 3*B[2*n+3,3])];
Table[A001797[n], {n,30}] (* G. C. Greubel, Apr 23 2025 *)
  • SageMath
    b=binomial
def A001797(n): return 20*b(n+1,2)*numerator(b(4*n,2*n)/2^(4*n))/(3*b(2*n+3,3))
print([A001797(n) for n in range(1,31)]) # G. C. Greubel, Apr 23 2025

Extensions

More terms from Sean A. Irvine, Nov 28 2012

A001798 Coefficients of Legendre polynomials.

Original entry on oeis.org

2, 28, 182, 4760, 31654, 428260, 2941470, 163761840, 1152562950, 16381761396, 117402623338, 3390322778024, 24634522766126, 360043025043380, 2644479279859438, 312191499849352032, 2312918756095439814, 34398444513178377492
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficient of Legendre_3(x) when x^n is written in term of Legendre polynomials. - Sean A. Irvine, Nov 28 2012

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001796.

Programs

  • Magma
    B:=Binomial;
A001798:= func< n | 14*B(n+2,3)*Numerator(B(4*n+2,2*n+1)/2^(4*n))/B(2*n+5,4) >;
[A001798(n): n in [1..30]]; // G. C. Greubel, Apr 23 2025
  • Maple
    a:=n->(14*n/((2*n+3)*(2*n+5)))*numer(binomial(4*n+2,2*n+1)/2^(4*n)); # Sean A. Irvine, Nov 28 2012
  • Mathematica
    A001798[n_]:= With[{B=Binomial}, 14*B[n+2,3]*Numerator[B[4*n+2,2*n+1]/2^(4*n) ]/B[2*n+5,4]];
Table[A001798[n], {n,30}] (* G. C. Greubel, Apr 23 2025 *)
  • SageMath
    b=binomial
def A001798(n): return 14*b(n+2,3)*numerator(b(4*n+2,2*n+1)/2^(4*n) )//b(2*n+5,4)
print([A001798(n) for n in range(1,31)]) # G. C. Greubel, Apr 23 2025

Formula

a(n) = (14*n/((2*n+3)*(2*n+5)))*numerator(binomial(4*n+2, 2*n+1)/2^(4*n)). - Sean A. Irvine, Nov 28 2012

Extensions

More terms from Sean A. Irvine, Nov 28 2012

A001799 Coefficients of Legendre polynomials.

Original entry on oeis.org

8, 72, 2160, 15504, 220248, 1564920, 89324640, 640807200, 9246847896, 67087213336, 1957095947664, 14342471475696, 211153052281080, 1560676296310488, 185256494416099008, 1379131680653181504, 20598677144877854232
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficient of Legendre_4(x) when x^n is written in term of Legendre polynomials. - Sean A. Irvine, Nov 28 2012

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001796.

Programs

  • Magma
    B:=Binomial;
A001799:= func< n | 144*B(n+2, 4)*Numerator(B(4*n, 2*n)/2^(4*n))/(5*B(2*n+5, 5)) >;
[A001799(n): n in [2..30]]; // G. C. Greubel, Apr 24 2025
  • Maple
    a:=n->(9*(2*n)*(2*n-2)/((2*n+1)*(2*n+3)*(2*n+5)))*numer(binomial(4*n,2*n)/2^(4*n)); # Sean A. Irvine, Nov 28 2012
  • Mathematica
    A001799[n_]:= With[{B=Binomial}, 144*B[n+2,4]*Numerator[B[4*n,2*n]/2^(4*n)]/(5*B[2*n+ 5, 5])];
Table[A001799[n], {n,2,35}] (* G. C. Greubel, Apr 24 2025 *)
  • SageMath
    b=binomial
def A001799(n): return 144*b(n+2, 4)*numerator(b(4*n, 2*n)/2^(4*n))//(5*b(2*n+5, 5))
print([A001799(n) for n in range(2, 31)]) # G. C. Greubel, Apr 24 2025

Formula

a(n) = (9*(2*n)*(2*n-2)/((2*n+1)*(2*n+3)*(2*n+5))) * numerator(binomial(4*n, 2*n)/2^(4*n)). - Sean A. Irvine, Nov 28 2012

Extensions

More terms from Sean A. Irvine, Nov 28 2012
Showing 1-6 of 6 results.

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

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