Real-rooted atlas

basis-map coefficients of A026729 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 16\,x + 16\,x^{2} + 12\,n x - 32\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 - 16\,x^{2} + 12\,n x + 48\,n x^{2} + 16\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
211617
3132144177
414843210241505
51648644096640011425
6180144010240320003686480625
719621602048096000221184200704540625
81112302435840224000774144140492810485763490625
91128403257344448000206438456197128388608530841621890625
101144518486016806400464486416859136377487364777574426214400134140625
11116064801228801344000928972842147840125829120238878720262144000126877696806640625
121176792016896021120001703116892725248346030080875888640144179200013956546566039797764775390625
13119295042252803168000291962881854504968304721922627665920576716800083739279367247757312283534950427900390625
14120811232292864457600047443968344408064179935641668319313921874329600036287021056471104225283685954355213153337344161181640625
151224131043727366406400738017286027141123598712832159411732485248122880012700457369621984863846425801680486418414672281660397977600922119140625
Technical details
Node id
n-c8692c346ff4c403 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6fb0656952daab5a

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  16;
   3: 1,  32,  144;
   4: 1,  48,  432,  1024;
   5: 1,  64,  864,  4096,   6400;
   6: 1,  80, 1440, 10240,  32000,   36864;
   7: 1,  96, 2160, 20480,  96000,  221184,   200704;
   8: 1, 112, 3024, 35840, 224000,  774144,  1404928,  1048576;
   9: 1, 128, 4032, 57344, 448000, 2064384,  5619712,  8388608,  5308416;
  10: 1, 144, 5184, 86016, 806400, 4644864, 16859136, 37748736, 47775744, 26214400;
%F Formula
The row polynomials satisfy (1+(12*n-16)*x+(16*n^2-32*n+16)*x^2)*P(n,x) = (1+12*n*x+(16*n^2+48*n-16)*x^2+64*n^2*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 12*n*x + (16*n^2 + 48*n - 16)*x^2 + 64*n^2*x^3)*P[n - 1, x])/(1 + (12*n - 16)*x + (16*n^2 - 32*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + 12*n*x + (16*n^2 + 48*n - 16)*x^2 + 64*n^2*x^3)*v[i - 1])/(1 + (12*n - 16)*x + (16*n^2 - 32*n + 16)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.