Real-rooted atlas

A038330 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + x + n\right)\,P_{n}(x) = \left(96 + 40\,x + 4\,x^{2} + 16\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 384 + 160\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
038416016560
1716846089606412800
2131072114688368645120256288000
32359296262144011468802457602560010246400000
4419430405662310431457280917504014745601228804096140800000
57381975041174405120792723456293601280642252808257536573440163843072000000
6128849018882362232012818790481920845571686423488102404110417924404019226214406553666560000000
722333829939246385646796842520176230422548578304076101451776169114337282466250752226492416117964802621441433600000000

All 15 rows

Technical details
Node id
n-d07e742c44ebf2a2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-76caa1d365acf455

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:            384,             160,              16;
  1:           7168,            4608,             960,              64;
  2:         131072,          114688,           36864,            5120,            256;
  3:        2359296,         2621440,         1146880,          245760,          25600,           1024;
  4:       41943040,        56623104,        31457280,         9175040,        1474560,         122880,          4096;
  5:      738197504,      1174405120,       792723456,       293601280,       64225280,        8257536,        573440,        16384;
  6:    12884901888,     23622320128,     18790481920,      8455716864,     2348810240,      411041792,      44040192,      2621440,       65536;
  7:   223338299392,    463856467968,    425201762304,    225485783040,    76101451776,    16911433728,    2466250752,    226492416,    11796480,     262144;
  8:  3848290697216,   8933531975680,   9277129359360,   5669356830720,  2254857830400,   608811614208,  112742891520,  14092861440,  1132462080,   52428800,   1048576;
  9: 65970697666560, 169324790677504, 196537703464960, 136064563937280, 62362925137920, 19842748907520, 4464618504192, 708669603840, 77510737920, 5536481280, 230686720, 4194304;
%F Formula
The row polynomials satisfy (n+5+x)*P(n,x) = (16*n+96+(4*n+40)*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 384 + 160*x + 16*x^2.
%t Mathematica
P[0, x_] := 384 + 160*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 96 + (4*n + 40)*x + 4*x^2)*P[n - 1, x])/(n + 5 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 384 + 160*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 96 + (4*n + 40)*x + 4*x^2)*v[i - 1])/(n + 5 + 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. [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.