Real-rooted atlas

A062264 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(4\,n + n^{2}\right)\,P_{n}(x) = \left(24 + 12\,x + 24\,n + 10\,n x + 7\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-8 - 12\,x - 4\,x^{2} - 4\,n - 6\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111516
2857215172
3575693273351576
43649563231927847013327
5223634180530600109201890126107704
6134245293640262035121680308704032210846712
77954551988525208708511884953950107623078543306538984
84673345131251201580964010638320435897011088001700161425649549898962
92729815585015125115441560894894004351347013711698278678435006424453715377631424
101588192535428424408197248157186015204039335301516394883834831064178406756754004010012841043912
11921406335342763888556943844655565955395354837483015448813384671166509767043013764465123623563063136521282493456
125334940545214503321603886492104041887248480298473575401477764288051974482561303679520230411610278220802162160960961820158924064187
13308450441551332629436452614397292003078970408802424197575801344591448205391312326415773650464335470707050980501053482000363854414232423801183932211336
141781539209658229334251601737504425475221935367472019128595285801174607546112528743170956176773218336439662443228063335280106808902298463456592191620563230608804131172992
Technical details
Node id
n-a8cc4406f37906f9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a9c2e1fe55724eee

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:       11,        5;
  2:       85,       72,        15;
  3:      575,      693,       273,       35;
  4:     3649,     5632,      3192,      784,       70;
  5:    22363,    41805,     30600,    10920,     1890,      126;
  6:   134245,   293640,    262035,   121680,    30870,     4032,     210;
  7:   795455,  1988525,   2087085,  1188495,   395010,    76230,    7854,    330;
  8:  4673345, 13125120,  15809640, 10638320,  4358970,  1108800,  170016,  14256,   495;
  9: 27298155, 85015125, 115441560, 89489400, 43513470, 13711698, 2786784, 350064, 24453, 715;
%F Formula
The row polynomials satisfy (n^2+4*n)*P(n,x) = (7*n^2+24*n+24+(3*n^2+10*n+12)*x)*P(n-1,x) - (4*n+8+(6*n+12)*x+(2*n+4)*x^2)*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[((7*n^2 + 24*n + 24 + (3*n^2 + 10*n + 12)*x)*P[n - 1, x] - (4*n + 8 + (6*n + 12)*x + (2*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];
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] = ((7*n^2 + 24*n + 24 + (3*n^2 + 10*n + 12)*x)*v[i - 1] - (4*n + 8 + (6*n + 12)*x + (2*n + 4)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.