Real-rooted atlas

A272867 → x ↦ x+1 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(13 + 16\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11316433
2169416360128161089
32197811212012908836967686435937
4285611406082947363427842415041054722790440962561185921
537129322848806195540973440098030406606336301632092160018048020480102439135393
64826809356441281185852722348153603079720802816686081841256968666726429157120684032010629129830440961291467969
76274851754060260821312218205092540544823502388895285066248132220096519936000025022215689021071362398914564564582458777604587521638442618442977

All 15 rows

Technical details
Node id
n-43ea2eaea8001298 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-befc9532e898cdfb

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:          13,           16,            4;
  2:         169,          416,          360,           128,            16;
  3:        2197,         8112,        12012,          9088,          3696,           768,            64;
  4:       28561,       140608,       294736,        342784,        241504,        105472,         27904,          4096,           256;
  5:      371293,      2284880,      6195540,       9734400,       9803040,       6606336,       3016320,        921600,        180480,         20480,          1024;
  6:     4826809,     35644128,    118585272,     234815360,     307972080,     281668608,     184125696,      86667264,      29157120,       6840320,       1062912,        98304,         4096;
  7:    62748517,    540602608,   2131221820,    5092540544,    8235023888,    9528506624,    8132220096,    5199360000,    2502221568,     902107136,     239891456,     45645824,      5877760,      458752,       16384;
  8:   815730721,   8031810176,  36606519456,  102464986624,  197060846528,  276001130496,  291115062784,  235821228032,  148247520768,   72560377856,   27561189376,   8040087552,   1766309888,   282591232,    31064064,    2097152,     65536;
  9: 10604499373, 117465223824, 607656638628, 1949876378112, 4347656868672, 7150848187392, 8988757290240, 8827521490944, 6864817669632, 4258530156544, 2112251590656, 835741679616, 261848186880, 64094994432, 11990532096, 1654652928, 158662656, 9437184, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (13+16*x+4*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[(13 + 16*x + 4*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] = (13 + 16*x + 4*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

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.