Real-rooted atlas

A139548 → basis A013609 → basis A013609

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1326432128
21024409661444096102416384
332768196608491520655360491520196608327682097152
410485768388608293601285872025673400320587202562936012883886081048576268435456
53355443233554432015099494404026531840704643072084557168647046430720402653184015099494403355443203355443234359738368
610737418241288490188870866960384236223201280531502202880850403524608992137445376850403524608531502202880236223201280708669603841288490188810737418244398046511104
7343597383684810363371523126736191488125069447659523439409810636868788196212736103182294319104117922622078976103182294319104687881962127363439409810636812506944765952312673619148848103633715234359738368562949953421312

All 15 rows

Technical details
Node id
n-d311e24793b27ad5 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-2cd9f46ca2d53115

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: yes
  • γ-positive (gamma_positive): yes
  • 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:             32,              64,               32;
  2:           1024,            4096,             6144,              4096,               1024;
  3:          32768,          196608,           491520,            655360,             491520,             196608,              32768;
  4:        1048576,         8388608,         29360128,          58720256,           73400320,           58720256,           29360128,             8388608,             1048576;
  5:       33554432,       335544320,       1509949440,        4026531840,         7046430720,         8455716864,         7046430720,          4026531840,          1509949440,           335544320,            33554432;
  6:     1073741824,     12884901888,      70866960384,      236223201280,       531502202880,       850403524608,       992137445376,        850403524608,        531502202880,        236223201280,         70866960384,         12884901888,         1073741824;
  7:    34359738368,    481036337152,    3126736191488,    12506944765952,     34394098106368,     68788196212736,    103182294319104,     117922622078976,     103182294319104,      68788196212736,      34394098106368,      12506944765952,      3126736191488,       481036337152,        34359738368;
  8:  1099511627776,  17592186044416,  131941395333120,   615726511554560,   2001111162552320,   4802666790125568,   8804889115230208,   12578413021757440,   14150714649477120,   12578413021757440,    8804889115230208,    4802666790125568,   2001111162552320,    615726511554560,    131941395333120,    17592186044416,    1099511627776;
  9: 35184372088832, 633318697598976, 5383208929591296, 28710447624486912, 107664178591825920, 301459700057112576, 653162683457077248, 1119707457354989568, 1539597753863110656, 1710664170959011840, 1539597753863110656, 1119707457354989568, 653162683457077248, 301459700057112576, 107664178591825920, 28710447624486912, 5383208929591296, 633318697598976, 35184372088832;
%F Formula
The row polynomials satisfy P(n,x) = (32+64*x+32*x^2)*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[(32 + 64*x + 32*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (32 + 64*x + 32*x^2)*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

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.