Real-rooted atlas

A272866 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(13 + 7\,x - 3\,x^{2} - 2\,x^{3} - 22\,n - 19\,n x - n x^{2} + 2\,n x^{3} + 9\,n^{2} + 12\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-4\,n - 7\,n x + 18\,n x^{2} + 28\,n x^{3} + 13\,n x^{4} + 2\,n x^{5} + 9\,n^{2} + 39\,n^{2} x + 49\,n^{2} x^{2} + 24\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1527
22858304120
3693304953007561275
4128980277236602460852140811000
520521709210202002491517970763018402251084375
6300406223130722401359601607701219265952018360339033012600000
7413681848825199164511980869484999229786520424872155820376535628455144046875

All 15 rows

Technical details
Node id
n-f3c39411cf79b61b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e0453429b9960a60

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   5,     2;
  2:  28,    58,     30,      4;
  3:  69,   330,    495,    300,      75,        6;
  4: 128,   980,   2772,   3660,    2460,      852,      140,        8;
  5: 205,  2170,   9210,  20200,   24915,    17970,     7630,     1840,      225,       10;
  6: 300,  4062,  23130,  72240,  135960,   160770,   121926,    59520,    18360,     3390,      330,      12;
  7: 413,  6818,  48825, 199164,  511980,   869484,   999229,   786520,   424872,   155820,    37653,    5628,     455,     14;
  8: 544, 10600,  91560, 462280, 1516200,  3401328,  5364800,  6040280,  4892040,  2853200,  1190112,  348600,   69160,   8680,    600,    16;
  9: 693, 15570, 157572, 949680, 3799530, 10657836, 21609000, 32233824, 35738415, 29603790, 18332424, 8447040, 2861586, 696780, 117180, 12672, 765, 18;
%F Formula
The row polynomials satisfy (9*n^2-22*n+13+(12*n^2-19*n+7)*x+(4*n^2-n-3)*x^2+(2*n-2)*x^3)*P(n,x) = (9*n^2-4*n+(39*n^2-7*n)*x+(49*n^2+18*n)*x^2+(24*n^2+28*n)*x^3+(4*n^2+13*n)*x^4+2*n*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 4*n + (39*n^2 - 7*n)*x + (49*n^2 + 18*n)*x^2 + (24*n^2 + 28*n)*x^3 + (4*n^2 + 13*n)*x^4 + 2*n*x^5)*P[n - 1, x])/(9*n^2 - 22*n + 13 + (12*n^2 - 19*n + 7)*x + (4*n^2 - n - 3)*x^2 + (2*n - 2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^2 - 4*n + (39*n^2 - 7*n)*x + (49*n^2 + 18*n)*x^2 + (24*n^2 + 28*n)*x^3 + (4*n^2 + 13*n)*x^4 + 2*n*x^5)*v[i - 1])/(9*n^2 - 22*n + 13 + (12*n^2 - 19*n + 7)*x + (4*n^2 - n - 3)*x^2 + (2*n - 2)*x^3)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.