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 15 results. Next

A111883 Unsigned row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).

Original entry on oeis.org

1, 1, 4, 16, 100, 676, 5776, 53824, 583696, 6864400, 90174016, 1274204416, 19642583104, 323196798016, 5714394630400, 107112895415296, 2135062451773696, 44858948563673344, 994634863541502976, 23133227941938073600, 564474119626559497216, 14388648533002088866816
Offset: 0

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

Links

Crossrefs

Cf. A111882 (row sums of A111595).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-x))/Sqrt(1-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018
  • Mathematica
    Table[Abs[HermiteH[n, I/Sqrt[2]]]^2/2^n, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 11 2016 *)
CoefficientList[Series[Exp[t/(1-t)]/Sqrt[1-t^2],{t,0,100}],t] Range[0, 12]! (* Emanuele Munarini, Aug 31 2017 *)
  • PARI
    a(n)=if(n<0, 0, n!*polcoeff(exp(x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)) /* Michael Somos, Aug 30 2005 */
  • Python
    from sympy import hermite, Poly, sqrt, I
def a(n): return abs(Poly(hermite(n, I/sqrt(2)), x))**2/2**n # Indranil Ghosh, May 26 2017

Formula

E.g.f.: exp(x/(1-x))/sqrt(1-x^2).
a(n) = A000085(n)^2. - Michael Somos, Aug 30 2005
Conjecture: a(n) -n*a(n-1) -n*(n-1)*a(n-2) +(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Oct 05 2014
Remark: the above conjectured recurrence is true and can be easily obtained by the e.g.f. - Emanuele Munarini, Aug 31 2017
a(n) = |H_n(i/sqrt(2))|^2 / 2^n = H_n(i/sqrt(2)) * H_n(-i/sqrt(2)) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1). - Vladimir Reshetnikov, Oct 11 2016
a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^n / 2. - Vaclav Kotesovec, Oct 01 2017

A111882 Row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).

Original entry on oeis.org

1, 1, 0, 4, 4, 36, 256, 400, 17424, 784, 1478656, 876096, 154753600, 560363584, 19057250304, 220388935936, 2564046397696, 83038749753600, 327933273309184, 33173161139160064, 26222822450021376, 14475245839622726656
Offset: 0

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

Links

Crossrefs

Cf. A111883 (unsigned row sums of A111595).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1+x))/Sqrt(1-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 10 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Exp[x/(1 + x)]/Sqrt[1 - x^2], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(x/(1+x))/sqrt(1-x^2))) \\ G. C. Greubel, Jun 10 2018
  • Python
    from sympy import hermite, Poly, sqrt
def a(n): return sum(Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()) # Indranil Ghosh, May 26 2017

Formula

E.g.f.: exp(x/(1+x))/sqrt(1-x^2).
a(n) = Sum_{m=0..n} A111595(n, m), n>=0.
A111882(n) = A001464(n)^2. - Mark van Hoeij, Nov 11 2009
D-finite with recurrence a(n) +(n-2)*a(n-1) -(n-1)*(n-2)*a(n-2) -(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Oct 05 2014

A112239 Matrix logarithm of triangle A111595.

Original entry on oeis.org

0, 0, 0, 1, -2, 0, 3, 3, -6, 0, 12, 12, 6, -12, 0, 60, 60, 30, 10, -20, 0, 360, 360, 180, 60, 15, -30, 0, 2520, 2520, 1260, 420, 105, 21, -42, 0, 20160, 20160, 10080, 3360, 840, 168, 28, -56, 0, 181440, 181440, 90720, 30240, 7560, 1512, 252, 36, -72, 0
Offset: 0

Views

Author

Paul D. Hanna, Aug 29 2005

Keywords

Comments

A111595 is the triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).

Examples

			Triangle begins:
0;
0,0;
1,-2,0;
3,3,-6,0;
12,12,6,-12,0;
60,60,30,10,-20,0;
360,360,180,60,15,-30,0;
2520,2520,1260,420,105,21,-42,0;
20160,20160,10080,3360,840,168,28,-56,0; ...

Crossrefs

Cf. A112239.

Programs

  • PARI
    T(n,k)=if(n<=k || k<0,0,if(n-1==k,-k*(k+1),n!/k!/2))

Formula

T(n, k) = n!/k!/2 for n-1>k>=0; T(k+1, k) = -k*(k+1), T(k, k) = 0 for k>=0.

A111602 Third column (m=2) of unsigned triangle A111595.

Original entry on oeis.org

1, 6, 42, 300, 2475, 22050, 220500, 2381400, 28279125, 360186750, 4970577150, 73045872900, 1150472498175, 19174541636250, 339663308985000, 6333077180430000, 124682456989715625, 2572020969902133750
Offset: 2

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

Formula

E.g.f.: (1/sqrt(1-x^2))*((x/(1-x))^2)/2!.
a(n) = (n!/2!)*Sum_{k=0..floor((n-2)/2)} binomial(2*k, k)*(n-2*k-1)/(4^k), n >= 2.
D-finite with recurrence (-n+2)*a(n) +2*n*a(n-1) +n*(n-1)^2*a(n-2)=0. - R. J. Mathar, Aug 11 2025

A111777 Fourth column (m=3) of unsigned triangle A111595.

Original entry on oeis.org

1, 12, 130, 1380, 15435, 182280, 2302020, 30958200, 444230325, 6771510900, 109568809350, 1874844071100, 33875023557375, 644264598978000, 12877256933541000, 269789087886318000, 5915648230774907625, 135459771081512377500, 3234745039813583546250
Offset: 3

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

Crossrefs

Cf. A111595.

Programs

  • PARI
    a(n) = {my(p = polhermite(n)^2/2^n); my(q = sum(k=1, poldegree(p), polcoef(p, k)/2^(k/2)*x^k)); abs(polcoef(q, 6));} \\ Michel Marcus, Jan 06 2021

Formula

E.g.f.: (1/sqrt(1-x^2))*((x/(1-x))^3)/3!.
a(n) = (n!/3!) * Sum_{k=0..floor((n-3)/2)} binomial(2*k, k)*binomial(n-2*k-1, 2)/(4^k), n >= 3.
D-finite with recurrence (-n+3)*a(n) +3*n*a(n-1) +n*(n-1)^2*a(n-2)=0. - R. J. Mathar, Aug 11 2025

A111784 Eleventh column (m=10) of unsigned triangle A111595.

Original entry on oeis.org

1, 110, 7326, 386100, 17846829, 762431670, 31039608600, 1227833727120, 47809764352350, 1849155516788580, 71501760198168300, 2777115998421765000, 108722966424618095550, 4301625967084096150500, 172338358130509601230200
Offset: 10

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

Programs

  • Mathematica
    With[{nn=30},Drop[CoefficientList[Series[1/Sqrt[1-x^2] (x/(1-x))^10/ 10!,{x,0,nn}],x] Range[0,nn]!,10]] (* Harvey P. Dale, May 22 2016 *)

Formula

E.g.f.: (1/sqrt(1-x^2))*((x/(1-x))^10)/10!.
a(n) = (n!/10!)*Sum_{k=0..floor((n-10)/2)} binomial(2*k, k)*binomial(n-2*k-1, 9)/(4^k), n >= 10.

A111778 Fifth column (m=4) of unsigned triangle A111595.

Original entry on oeis.org

1, 20, 315, 4620, 67830, 1020600, 15961050, 260706600, 4461170175, 80002622700, 1503527550525, 29583578524500, 608837350621500, 13088359506222000, 293538127312930500, 6858722586405690000, 166752692881988338125
Offset: 4

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

Programs

  • Mathematica
    With[{nn=30},Drop[CoefficientList[Series[1/Sqrt[1-x^2] (x/(1-x))^4/4!,{x,0,nn}],x] Range[0,nn]!,4]] (* Harvey P. Dale, Apr 05 2015 *)

Formula

E.g.f. (1/sqrt(1-x^2))*((x/(1-x))^4)/4!.
a(n)=(n!/4!)*sum(binomial(2*k, k)*binomial(n-2*k-1, 3)/(4^k), k=0..floor((n-4)/2)), n>=4.
D-finite with recurrence (-n+4)*a(n) +4*n*a(n-1) +n*(n-1)^2*a(n-2)=0. - R. J. Mathar, Jun 08 2016

A111779 Sixth column (m=5) of unsigned triangle A111595.

Original entry on oeis.org

1, 30, 651, 12600, 235494, 4396140, 83471850, 1627358040, 32754696975, 682573101210, 14749179169725, 330654317994000, 7691170398011100, 185559161386599000, 4641068950134516900, 120256269348313098000
Offset: 5

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

Formula

E.g.f. (1/sqrt(1-x^2))*((x/(1-x))^5)/5!.
a(n)=(n!/5!)*sum(binomial(2*k, k)*binomial(n-2*k-1, 4)/(4^k), k=0..floor((n-5)/2)), n>=5.
Conjecture: +(-n+5)*a(n) +5*n*a(n-1) +n*(n-1)^2*a(n-2)=0. - R. J. Mathar, Jun 08 2016

A111780 Seventh column (m=6) of unsigned triangle A111595.

Original entry on oeis.org

1, 42, 1204, 29736, 689850, 15647940, 354718980, 8137289160, 190349674515, 4561677870750, 112317990384600, 2846259737521200, 74306186469414900, 1999421553976446600, 55458660913310655000, 1585528821898619598000
Offset: 6

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

Formula

E.g.f. (1/sqrt(1-x^2))*((x/(1-x))^6)/6!.
a(n)=(n!/6!)*sum(binomial(2*k, k)*binomial(n-2*k-1, 5)/(4^k), k=0..floor((n-6)/2)), n>=6.

A111781 Eighth column (m=7) of unsigned triangle A111595.

Original entry on oeis.org

1, 56, 2052, 63000, 1777050, 48149640, 1284709140, 34260506280, 921799753875, 25175488338000, 700755564108600, 19932553764723600, 580406741946731700, 17320767850295910000, 530112107596146075000, 16645556950225160958000
Offset: 7

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

Formula

E.g.f. (1/sqrt(1-x^2))*((x/(1-x))^7)/7!.
a(n)=(n!/7!)*sum(binomial(2*k, k)*binomial(n-2*k-1, 6)/(4^k), k=0..floor((n-7)/2)), n>=7.
Showing 1-10 of 15 results. Next

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