Real-rooted atlas

A034870 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 - 2\,x + 4\,n x\right)\,P_{n}(x) = \left(4 + 24\,x + 48\,x^{2} + 32\,x^{3} + 16\,n x + 64\,n x^{2} + 64\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14324884
2162561152204812804752
36415361152040960768007372828672233280
4256819286016458752143360027525123211264209715258982410637568
510244096055296039321601720320049545216963379201258291201061683205242880011534336463574016
6409619660832440322883584016220160062285414416955473923321888768467140608046137344003045064704120795955221810380819591041024
7163849175041789132819084083213120307206297747456220421160965757940531211335945420816793993216018473392537614656575897679389786112263066746884026531840809763028992

All 15 rows

Technical details
Node id
n-a228635cac40c96b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-111fd8abade84f64

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      4,       32,        48;
  2:     16,      256,      1152,       2048,        1280;
  3:     64,     1536,     11520,      40960,       76800,        73728,         28672;
  4:    256,     8192,     86016,     458752,     1433600,      2752512,       3211264,       2097152,         589824;
  5:   1024,    40960,    552960,    3932160,    17203200,     49545216,      96337920,     125829120,      106168320,       52428800,        11534336;
  6:   4096,   196608,   3244032,   28835840,   162201600,    622854144,    1695547392,    3321888768,     4671406080,     4613734400,      3045064704,      1207959552,       218103808;
  7:  16384,   917504,  17891328,  190840832,  1312030720,   6297747456,   22042116096,   57579405312,   113359454208,   167939932160,    184733925376,    146565758976,     79389786112,     26306674688,      4026531840;
  8:  65536,  4194304,  94371840, 1174405120,  9542041600,  54962159616,  235115905024,  767725404160,  1943304929280,  3838627020800,   5911485612032,   7035156430848,   6351182888960,   4209067950080,   1932735283200,    549755813888,    73014444032;
  9: 262144, 18874368, 481296384, 6845104128, 64172851200, 431241560064, 2180165664768, 8542689951744, 26428947038208, 65256659353600, 129208185520128, 205024558841856, 259128261869568, 257594958544896, 197138998886400, 112150186033152, 44684839747584, 11132555231232, 1305670057984;
%F Formula
The row polynomials satisfy (1+(4*n-2)*x)*P(n,x) = (4+(16*n+24)*x+(64*n+48)*x^2+(64*n+32)*x^3)*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[((4 + (16*n + 24)*x + (64*n + 48)*x^2 + (64*n + 32)*x^3)*P[n - 1, x])/(1 + (4*n - 2)*x)]];
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] = ((4 + (16*n + 24)*x + (64*n + 48)*x^2 + (64*n + 32)*x^3)*v[i - 1])/(1 + (4*n - 2)*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

Same fingerprint ?

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.