Real-rooted atlas

A075497 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{4}\right)\,P_{n-4}(x) + \left(2\,x^{4}\right)\,P_{n-4}'(x) + \left(8\,x + 2\,x^{2} - 2\,x^{3}\right)\,P_{n-3}(x) + \left(4\,x^{2} - 2\,x^{3}\right)\,P_{n-3}'(x) + \left(-8 - 8\,x\right)\,P_{n-2}(x) + \left(-4\,x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(6 + 2\,x\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
\(P_{2}(x) = 16 + 18\,x + 3\,x^{2}\)
\(P_{3}(x) = 64 + 128\,x + 48\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21618337
364128484244
425684054010051741
51024531252801640180613442
6409632928480482240040602947111833
7163842017284193282768647392087364488997416
86553612268803564480321888011911202046241696864899489145
926214474270722977536035946880177609604167072497280304809001095868158
1010485764482508824576076839043840025126112077460768124887841092960514801210111024429165
11419430427000012820114350084157441024342422784013515025922830248963318585622176008272015841211537313564
12167772161624197120163635763204363014656045404719360225218273285964614656900446976800822884215640127556202813136510733061
1367108864976196403213254119424045293294592058978343424036263247820811914878976022538447360256288032017857840075915841900082548141692155605498
1426843545658638901248107012506828846634864640007540975303680568715654707222852528258565311549900807513444224066559292803728204481306032027482031501521919235065953
Technical details
Node id
n-b41c378bbf9ce01d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3bfcfaad6895e72d

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:      4,       2;
  2:     16,      18,        3;
  3:     64,     128,       48,        4;
  4:    256,     840,      540,      100,        5;
  5:   1024,    5312,     5280,     1640,      180,       6;
  6:   4096,   32928,    48048,    22400,     4060,     294,      7;
  7:  16384,  201728,   419328,   276864,    73920,    8736,    448,     8;
  8:  65536, 1226880,  3564480,  3218880,  1191120,  204624,  16968,   648,   9;
  9: 262144, 7427072, 29775360, 35946880, 17760960, 4167072, 497280, 30480, 900, 10;
%F Formula
The row polynomials satisfy P(n,x) = x^4*P(n-4,x) + 2*x^4*d/dx P(n-4,x) + (8*x+2*x^2-2*x^3)*P(n-3,x) + (4*x^2-2*x^3)*d/dx P(n-3,x) - (8+8*x)*P(n-2,x) - (4*x+2*x^2)*d/dx P(n-2,x) + (6+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 16 + 18*x + 3*x^2, P(3,x) = 64 + 128*x + 48*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[2, x_] := 16 + 18*x + 3*x^2;
P[3, x_] := 64 + 128*x + 48*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[x^4*P[n - 4, x] + 2*x^4*D[P[n - 4, x], x] + (8*x + 2*x^2 - 2*x^3)*P[n - 3, x] + (4*x^2 - 2*x^3)*D[P[n - 3, x], x] - (8 + 8*x)*P[n - 2, x] - (4*x + 2*x^2)*D[P[n - 2, x], x] + (6 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], 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] = 4 + 2*x; v[3] = 16 + 18*x + 3*x^2; v[4] = 64 + 128*x + 48*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = x^4*v[i - 4] + 2*x^4*deriv(v[i - 4], x) + (8*x + 2*x^2 - 2*x^3)*v[i - 3] + (4*x^2 - 2*x^3)*deriv(v[i - 3], x) - (8 + 8*x)*v[i - 2] - (4*x + 2*x^2)*deriv(v[i - 2], x) + (6 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.