Real-rooted atlas

A079638 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-48 + 112\,n - 92\,n^{2} + 32\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(-36 + 8\,x + 78\,n - 12\,n x - 54\,n^{2} + 4\,n^{2} x + 12\,n^{3}\right)\,P_{n-3}(x) + \left(-13 + 6\,x - x^{2} + 26\,n - 6\,n x - 13\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 14 + 18\,x + 3\,x^{2}\)
\(P_{3}(x) = 90 + 166\,x + 54\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21418335
390166544314
4744174082512053434
5756020948128252740225644304
6914402877842127725922071753787658776
7128520044667123822588129715620580016044588811094096
8206035207753968074611716295958885757045589680320468649208730448
937122624014915718721580373180711780800162799875205194781468530588001215104339800000
107428153600315324662403621844432818133328400473745415070480179062227011328680010098016501198820264960
1116345929600072728188128089482526258449009932030414318667495024429118890255268351816664498467626901644502178122446007811840
123923502105600181844952307202374296297235214049477183456451665655298086520461742010396931444980488568164042737271299870025639928081365394734365440
1310201728176640049013037124992067406681684160042667590618307214903720946114031541082849276426777139288538073425447222711679991904833930236185953854763549141878140814520320
142856571067750400141703538920473602040356334228595213705134120699840514884469481200011886356502820801781994980325561793850782724012350609592755850669825018939690774093362056192544101557670536276864000
Technical details
Node id
n-e6b32969c42863c1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3b33a02937f0bff7

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         3,          2;
  2:        14,         18,          3;
  3:        90,        166,         54,         4;
  4:       744,       1740,        825,       120,         5;
  5:      7560,      20948,      12825,      2740,       225,        6;
  6:     91440,     287784,     212772,     59220,      7175,      378,       7;
  7:   1285200,    4466712,    3822588,   1297156,    205800,    16044,     588,     8;
  8:  20603520,   77539680,   74611716,  29595888,   5757045,   589680,   32046,   864,    9;
  9: 371226240, 1491571872, 1580373180, 711780800, 162799875, 20519478, 1468530, 58800, 1215, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n^4-32*n^3+92*n^2-112*n+48)*P(n-4,x) + (12*n^3-54*n^2+78*n-36+(4*n^2-12*n+8)*x)*P(n-3,x) - (13*n^2-26*n+13+(6*n-6)*x+x^2)*P(n-2,x) + (6*n-3+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 14 + 18*x + 3*x^2, P(3,x) = 90 + 166*x + 54*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 14 + 18*x + 3*x^2;
P[3, x_] := 90 + 166*x + 54*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(4*n^4 - 32*n^3 + 92*n^2 - 112*n + 48)*P[n - 4, x] + (12*n^3 - 54*n^2 + 78*n - 36 + (4*n^2 - 12*n + 8)*x)*P[n - 3, x] - (13*n^2 - 26*n + 13 + (6*n - 6)*x + x^2)*P[n - 2, x] + (6*n - 3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 14 + 18*x + 3*x^2; v[4] = 90 + 166*x + 54*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 32*n^3 + 92*n^2 - 112*n + 48)*v[i - 4] + (12*n^3 - 54*n^2 + 78*n - 36 + (4*n^2 - 12*n + 8)*x)*v[i - 3] - (13*n^2 - 26*n + 13 + (6*n - 6)*x + x^2)*v[i - 2] + (6*n - 3 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023