Real-rooted atlas

basis-map coefficients of A297191, shifted → (θ+1)² → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19363681
2252006008004002025
34958829407840117609408313635721
4811296907236288907201451521451528294420736531441
51212420217801161604065609757441626240185856013939206195201239047144929
616940564461629744013384804283136999398417132544214156801903616011421696415334469222489813529
72256300819006552003603600144144004324320098841600172972800230630400230630400167731200838656002580480036864001076168025

All 15 rows

Technical details
Node id
n-dc9add5d99860f64 · revision 0 · JSON
Definition source
derived:basis_map_A013609
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:   9,    36,     36;
  2:  25,   200,    600,     800,      400;
  3:  49,   588,   2940,    7840,    11760,     9408,      3136;
  4:  81,  1296,   9072,   36288,    90720,   145152,    145152,      82944,      20736;
  5: 121,  2420,  21780,  116160,   406560,   975744,   1626240,    1858560,    1393920,     619520,      123904;
  6: 169,  4056,  44616,  297440,  1338480,  4283136,   9993984,   17132544,   21415680,   19036160,    11421696,     4153344,      692224;
  7: 225,  6300,  81900,  655200,  3603600, 14414400,  43243200,   98841600,  172972800,  230630400,   230630400,   167731200,    83865600,    25804800,     3686400;
  8: 289,  9248, 138720, 1294720,  8415680, 40395264, 148115968,  423188480,  952174080, 1692753920,  2369855488,  2585296896,  2154414080,  1325793280,   568197120,  151519232,   18939904;
  9: 361, 12996, 220932, 2356608, 17674560, 98977536, 428902656, 1470523392, 4043939328, 8986531840, 16175757312, 23528374272, 27449769984, 25338249216, 18098749440, 9652666368, 3619749888, 851705856, 94633984;
%F Formula
The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (4*n^2+4*n+1+(16*n^2+16*n+4)*x+(16*n^2+16*n+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[Together[((4*n^2 + 4*n + 1 + (16*n^2 + 16*n + 4)*x + (16*n^2 + 16*n + 4)*x^2)*P[n - 1, x])/(4*n^2 - 4*n + 1)]];
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*n^2 + 4*n + 1 + (16*n^2 + 16*n + 4)*x + (16*n^2 + 16*n + 4)*x^2)*v[i - 1])/(4*n^2 - 4*n + 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.