Real-rooted atlas

A034870 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(4 + 32\,x + 88\,x^{2} + 96\,x^{3} + 36\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14243664
216192864172812964096
3641152864034560777609331246656262144
42566144645123870721451520348364852254724478976167961616777216
510243072041472033177601741824062705664156764160268738560302330880201553920604661761073741824
64096147456243302424330240164229120788299776275904921670946979841330255872017736744960159630704648707129344217678233668719476736
7163846881281341849616102195213284311047970586624358676398081229747650563228087582726456175165449684262748161056465027072792348770304365699432448783641640964398046511104

All 15 rows

Technical details
Node id
n-db2d0b2ce9624a44 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-12929af9fbcc0757

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,       24,        36;
  2:     16,      192,       864,       1728,        1296;
  3:     64,     1152,      8640,      34560,       77760,        93312,         46656;
  4:    256,     6144,     64512,     387072,     1451520,      3483648,       5225472,        4478976,        1679616;
  5:   1024,    30720,    414720,    3317760,    17418240,     62705664,     156764160,      268738560,      302330880,       201553920,        60466176;
  6:   4096,   147456,   2433024,   24330240,   164229120,    788299776,    2759049216,     7094697984,    13302558720,     17736744960,     15963070464,       8707129344,       2176782336;
  7:  16384,   688128,  13418496,  161021952,  1328431104,   7970586624,   35867639808,   122974765056,   322808758272,    645617516544,    968426274816,    1056465027072,     792348770304,     365699432448,      78364164096;
  8:  65536,  3145728,  70778880,  990904320,  9661317120,  69561483264,  382588157952,  1639663534080,  5533864427520,  14756971806720,  30989640794112,   50710321299456,   63387901624320,   58511909191680,   37614798766080,   15045919506432,    2821109907456;
  9: 262144, 14155776, 360972288, 5775556608, 64975011840, 545790099456, 3547635646464, 18244983324672, 75260556214272, 250868520714240, 677345005928448, 1477843649298432, 2586226386272256, 3580928842530816, 3836709474140160, 3069367579312128, 1726519263363072, 609359740010496, 101559956668416;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (4+32*x+88*x^2+96*x^3+36*x^4)*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 + 32*x + 88*x^2 + 96*x^3 + 36*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
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 + 32*x + 88*x^2 + 96*x^3 + 36*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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.