Real-rooted atlas

A126216 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-24 - 54\,x + 54\,x^{2} + 6\,n + 81\,n x + 27\,n x^{2} + 63\,n^{2} + 54\,n^{2} x + 27\,n^{3}\right)\,P_{n}(x) = \left(-24 - 54\,x + 54\,x^{2} - 12\,n - 108\,n x + 189\,n x^{2} + 243\,n x^{3} + 279\,n^{2} + 621\,n^{2} x + 486\,n^{2} x^{2} + 189\,n^{3} + 243\,n^{3} x\right)\,P_{n-1}(x) + \left(-20 - 126\,x - 288\,x^{2} - 324\,x^{3} - 162\,x^{4} - 66\,n - 369\,n x - 621\,n x^{2} - 324\,n x^{3} - 36\,n^{2} - 162\,n^{2} x - 162\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
23239980
317134218927729
492325982394810816806
55028183452412013770324024364746
6275981240292135701759057047012393729624694
71524218150521742202189567011056503333964592721876092505
8846101525082813435128182694961418180462732881490076166212656159919494
9471673833339087994709161627743601588461849411973233006204637729259049019683593260686
1026388588209351475714158235136856061016147933201209145644571178790163828170263752202066715590495905905816
111480875031303276194500586817511006269065152667593101388101730483253097623232539090776199105106091370714492917714759058738954
12833233551805711768234421844402854633567701364462921251463571105481069994736365297033847617293975875353998755041700403824446286531441592824712380
13469908921649528455093233014911948645067254546116609151344414438794325671249312727520756549251172316121104728883271278891563830096416074318788290479615943235969921100084
14265544994183030239796571556954973756475643130007596088326073741350481803570913525344582750972231238192549853698794901788905102985433853079660672380924676093502506279006525478296960286015807266
Technical details
Node id
n-2ce5f5474899f519 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bc83e7685d3c0684

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:       6,        3;
  2:      32,       39,        9;
  3:     171,      342,      189,        27;
  4:     923,     2598,     2394,       810,        81;
  5:    5028,    18345,    24120,     13770,      3240,      243;
  6:   27598,   124029,   213570,    175905,     70470,    12393,      729;
  7:  152421,   815052,  1742202,   1895670,   1105650,   333396,    45927,    2187;
  8:  846101,  5250828, 13435128,  18269496,  14181804,  6273288,  1490076,  166212,   6561;
  9: 4716738, 33339087, 99470916, 162774360, 158846184, 94119732, 33006204, 6377292, 590490, 19683;
%F Formula
The row polynomials satisfy (27*n^3+63*n^2+6*n-24+(54*n^2+81*n-54)*x+(27*n+54)*x^2)*P(n,x) = (189*n^3+279*n^2-12*n-24+(243*n^3+621*n^2-108*n-54)*x+(486*n^2+189*n+54)*x^2+243*n*x^3)*P(n-1,x) - (36*n^2+66*n+20+(162*n^2+369*n+126)*x+(162*n^2+621*n+288)*x^2+(324*n+324)*x^3+162*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((189*n^3 + 279*n^2 - 12*n - 24 + (243*n^3 + 621*n^2 - 108*n - 54)*x + (486*n^2 + 189*n + 54)*x^2 + 243*n*x^3)*P[n - 1, x] - (36*n^2 + 66*n + 20 + (162*n^2 + 369*n + 126)*x + (162*n^2 + 621*n + 288)*x^2 + (324*n + 324)*x^3 + 162*x^4)*D[P[n - 1, x], x])/(27*n^3 + 63*n^2 + 6*n - 24 + (54*n^2 + 81*n - 54)*x + (27*n + 54)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((189*n^3 + 279*n^2 - 12*n - 24 + (243*n^3 + 621*n^2 - 108*n - 54)*x + (486*n^2 + 189*n + 54)*x^2 + 243*n*x^3)*v[i - 1] - (36*n^2 + 66*n + 20 + (162*n^2 + 369*n + 126)*x + (162*n^2 + 621*n + 288)*x^2 + (324*n + 324)*x^3 + 162*x^4)*deriv(v[i - 1], x))/(27*n^3 + 63*n^2 + 6*n - 24 + (54*n^2 + 81*n - 54)*x + (27*n + 54)*x^2)); 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

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