Real-rooted atlas

A137320 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x^{2} - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
210818
3527224148
4308568336641276
5208846403720128016011888
6160564083239960188804320384120432
713852839088044217625760081760134408961325280
81327392407203251522243497536137984031718439424204815787680
91399968046029888637383364871462422542240648345611343361105924608202757760
10161254080562327488839211360705829760369751200125419392276057603809280299520102402795518080
1120148825607392946176117575187841070785408062015888002392435584623914368108810240121651207884802252841212926720
12271514764801041678305281750265827201705554864641073902896004598547571213693236480284062310440296960037283840202752049152647253331200
13392574712320156695553331227628756496642854514364416192985277056090153335500829900475404871067549696120329763841417461760110409728511180810649610791944755200
14606297726720025074631848960461440013448965018538386534436084077469440181546469744646588649139072174945268326434039663257648001873920477456179231757107212673024229376190437324134400
Technical details
Node id
n-f770ce833b7841d5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8f92721e6baca82c

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        2;
  2:       10,        8;
  3:       52,       72,       24;
  4:      308,      568,      336,       64;
  5:     2088,     4640,     3720,     1280,      160;
  6:    16056,    40832,    39960,    18880,     4320,     384;
  7:   138528,   390880,   442176,   257600,    81760,   13440,     896;
  8:  1327392,  4072032,  5152224,  3497536,  1379840,  317184,   39424,   2048;
  9: 13999680, 46029888, 63738336, 48714624, 22542240, 6483456, 1134336, 110592, 4608;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+4*n*x+4*x^2)*P(n-2,x) + (2*n+1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n*x + 4*x^2)*P[n - 2, x] + (2*n + 1 + 4*x)*P[n - 1, x]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n*x + 4*x^2)*v[i - 2] + (2*n + 1 + 4*x)*v[i - 1]); 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.