Real-rooted atlas

basis-map coefficients of A112227 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 8\,x + 16\,x^{2} + 12\,n x + 32\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 24\,x + 144\,x^{2} + 256\,x^{3} + 12\,n x + 112\,n x^{2} + 256\,n x^{3} + 16\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1 + 32\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
1132144177
214843210241505
31648644096640011425
4180144010240320003686480625
519621602048096000221184200704540625
61112302435840224000774144140492810485763490625
71128403257344448000206438456197128388608530841621890625
81144518486016806400464486416859136377487364777574426214400134140625
9116064801228801344000928972842147840125829120238878720262144000126877696806640625
101176792016896021120001703116892725248346030080875888640144179200013956546566039797764775390625
11119295042252803168000291962881854504968304721922627665920576716800083739279367247757312283534950427900390625
12120811232292864457600047443968344408064179935641668319313921874329600036287021056471104225283685954355213153337344161181640625
131224131043727366406400738017286027141123598712832159411732485248122880012700457369621984863846425801680486418414672281660397977600922119140625
1412401512046592087360001107025921004523520674758656034159656960131203072000381013721088824432394240129008402432013811004211209059696640002748779069445230712890625
151256172805734401164800016102195216072376321199570944068319313920299892736000101603658956826381836615685160336097280736586891264072477573120004398046511104124124554854429449462890625
Technical details
Node id
n-428df32f2c9ad121 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6fb0656952daab5a

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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,  32,  144;
   2: 1,  48,  432,   1024;
   3: 1,  64,  864,   4096,    6400;
   4: 1,  80, 1440,  10240,   32000,    36864;
   5: 1,  96, 2160,  20480,   96000,   221184,   200704;
   6: 1, 112, 3024,  35840,  224000,   774144,  1404928,   1048576;
   7: 1, 128, 4032,  57344,  448000,  2064384,  5619712,   8388608,   5308416;
   8: 1, 144, 5184,  86016,  806400,  4644864, 16859136,  37748736,  47775744,   26214400;
   9: 1, 160, 6480, 122880, 1344000,  9289728, 42147840, 125829120, 238878720,  262144000,  126877696;
  10: 1, 176, 7920, 168960, 2112000, 17031168, 92725248, 346030080, 875888640, 1441792000, 1395654656, 603979776;
%F Formula
The row polynomials satisfy (1+(12*n+8)*x+(16*n^2+32*n+16)*x^2)*P(n,x) = (1+(12*n+24)*x+(16*n^2+112*n+144)*x^2+(64*n^2+256*n+256)*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 32*x + 144*x^2.
%t Mathematica
P[1, x_] := 1 + 32*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (12*n + 24)*x + (16*n^2 + 112*n + 144)*x^2 + (64*n^2 + 256*n + 256)*x^3)*P[n - 1, x])/(1 + (12*n + 8)*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 + 32*x + 144*x^2; for(n = 2, N, my(i = n); v[i] = ((1 + (12*n + 24)*x + (16*n^2 + 112*n + 144)*x^2 + (64*n^2 + 256*n + 256)*x^3)*v[i - 1])/(1 + (12*n + 8)*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.