Real-rooted atlas

A166340 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n\right)\,P_{n}(x) = \left(-4\,x + 6\,x^{2} - 2\,x^{3} - 2\,n + 3\,n x - 5\,n x^{2} + 4\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x + 2\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 4\,x^{3} - 4\,x^{4} + 2\,x^{5} - n x + 2\,n x^{2} - 4\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 7\,x - 4\,x^{2} + 2\,x^{3} + 3\,n - 3\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 3\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 9 + 10\,x + x^{2}\)
\(P_{2}(x) = 20 + 57\,x + 22\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
1910120
22057221100
343270240461600
490110920408669414200
5185413414215119802895190133600
637614433856141309756134092793821302400
77594818246457411947549967202892262901876613024000
81526155925233860095196961314884467142741294200894001534133264000
93061493710111437256856868014772241012370150841663570559540027286530701399168000
10613215392815098395045766812514673518401899967426104304966824414525023638300827885614215189184000
1112275474660622623391628818757501325382091525310010620213727454088124908508137341362598293910250218012286172648576000
1224562145221339808322641734568532611114893649030147526559137346204822421771259141259676333798750056763540426447275435182457411089728640000
134913744180438417735585910077010858087897260440132859524123065747656532139490042695247220566152895954192877998984008231705716498845482270441949150117435658240000
149828813386800117548606078569236942955663215444713233353666615705798791520941789583275136149958731192801720183251717448912847637086306210779310521669674354868260875983021296406190080000
15196591404454342729381777543144310363530481789739206663195528216970141021951252631882167193436667988214391586067860806553987285018581559565452284141883998657515010952057986142707515389020507772619660615335311421440000
Technical details
Node id
n-f5d02bfef37a45e0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f3151dedff6a1901

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    9,      10,        1;
   2:   20,      57,       22,         1;
   3:   43,     270,      240,        46,          1;
   4:   90,    1109,     2040,       866,         94,          1;
   5:  185,    4134,    14215,     11980,       2895,        190,          1;
   6:  376,   14433,    85614,    130975,      61340,       9279,        382,         1;
   7:  759,   48182,   464574,   1194754,     996720,     289226,      29018,       766,        1;
   8: 1526,  155925,  2338600,   9519696,   13148844,    6714274,    1294200,     89400,     1534,      1;
   9: 3061,  493710, 11143725,  68568680,  147722410,  123701508,   41663570,   5595400,   272865,   3070,    1;
  10: 6132, 1539281, 50983950, 457668125, 1467351840, 1899967426, 1043049668, 244145250, 23638300, 827885, 6142, 1;
%F Formula
The row polynomials satisfy (n-1+2*x)*P(n,x) = -(2*n+(2*n^2-3*n+4)*x+(n^3-2*n^2+5*n-6)*x^2+(3*n^2-4*n+2)*x^3+2*n*x^4)*P(n-2,x) - (n*x+(n^2-2*n+2)*x^2-(n^2-4*n+4)*x^3-(3*n-4)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-2+(2*n^2-3*n+7)*x+(5*n-4)*x^2+2*x^3)*P(n-1,x) + ((n-1)*x-(n-3)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 9 + 10*x + x^2, P(2,x) = 20 + 57*x + 22*x^2 + x^3.
%t Mathematica
P[1, x_] := 9 + 10*x + x^2;
P[2, x_] := 20 + 57*x + 22*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*P[n - 2, x] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 9 + 10*x + x^2; v[2] = 20 + 57*x + 22*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*v[i - 2] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*v[i - 1] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 1 + 2*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.