Real-rooted atlas

A139548 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(32 + 32\,x + 8\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13232872
2102420481536512645184
3327689830412288081920307206144512373248
410485764194304734003273400324587520183500845875265536409626873856
533554432167772160377487360503316480440401920264241152110100480314572805898240655360327681934917632
610737418246442450944177167400962952790016033218887680265751101441550214758466437775362076180480461373440692060166291456262144139314069504
734359738368240518168576781684047872156336809574421496311316482149631131648161222334873692127048499240305583718413435194572833587986432610690662476336332858720256209715210030613004288

All 15 rows

Technical details
Node id
n-4d8246302789a4f5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-90334921cba1a1bb

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:             32,              32,                8;
  2:           1024,            2048,             1536,              512,               64;
  3:          32768,           98304,           122880,            81920,            30720,             6144,               512;
  4:        1048576,         4194304,          7340032,          7340032,          4587520,          1835008,            458752,            65536,             4096;
  5:       33554432,       167772160,        377487360,        503316480,        440401920,        264241152,         110100480,         31457280,          5898240,           655360,            32768;
  6:     1073741824,      6442450944,      17716740096,      29527900160,      33218887680,      26575110144,       15502147584,       6643777536,       2076180480,        461373440,         69206016,         6291456,          262144;
  7:    34359738368,    240518168576,     781684047872,    1563368095744,    2149631131648,    2149631131648,     1612223348736,     921270484992,     403055837184,     134351945728,      33587986432,      6106906624,       763363328,       58720256,       2097152;
  8:  1099511627776,   8796093022208,   32985348833280,   76965813944320,  125069447659520,  150083337191424,   137576392425472,   98268851732480,   55276229099520,   24567212933120,    8598524526592,   2345052143616,    488552529920,    75161927680,    8053063680,    536870912,    16777216;
  9: 35184372088832, 316659348799488, 1345802232397824, 3588805953060864, 6729011161989120, 9420615626784768, 10205666929016832, 8747714510585856, 6014053726027776, 3341140958904320, 1503513431506944, 546732156911616, 159463545765888, 36799279792128, 6571299962880, 876173328384, 82141249536, 4831838208, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (32+32*x+8*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 + 32*x + 8*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 + 32*x + 8*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.