Real-rooted atlas

A272867 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 36\,x - 36\,x^{2} - 6\,x^{3} + x^{4} + 108\,n x + 48\,n x^{2} - 6\,n x^{3} - 4\,n x^{4} + 36\,n^{2} x^{2} + 24\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(216 + 648\,x + 756\,x^{2} + 432\,x^{3} + 126\,x^{4} + 18\,x^{5} + x^{6} + 648\,n x + 1368\,n x^{2} + 1080\,n x^{3} + 396\,n x^{4} + 66\,n x^{5} + 4\,n x^{6} + 216\,n^{2} x^{2} + 360\,n^{2} x^{3} + 204\,n^{2} x^{4} + 48\,n^{2} x^{5} + 4\,n^{2} x^{6}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1624939
23628843219225973
321625926804691231506484920371
4129620736777601244161026004665611760153681386841
577761555207581601658880199800014463366526801843203159030001216896383
646656111974467184641866240029646000296732161964390487920642668140540000696965184169117585637
7279936783820855847232188116992371498400476171136418300848259448832115246152367416008324316130636813486282322251939263339

All 15 rows

Technical details
Node id
n-cdbfc3bc09489b9a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-19a74c9c27488ed0

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,        24,          9;
  2:       36,       288,        432,         192,          25;
  3:      216,      2592,       6804,        6912,        3150,         648,           49;
  4:     1296,     20736,      77760,      124416,      102600,       46656,        11760,         1536,          81;
  5:     7776,    155520,     758160,     1658880,     1998000,     1446336,       652680,       184320,       31590,        3000,         121;
  6:    46656,   1119744,    6718464,    18662400,    29646000,    29673216,     19643904,      8792064,     2668140,      540000,       69696,       5184,        169;
  7:   279936,   7838208,   55847232,   188116992,   371498400,   476171136,    418300848,    259448832,   115246152,    36741600,     8324316,    1306368,     134862,      8232,      225;
  8:  1679616,  53747712,  443418624,  1755758592,  4147718400,  6490036224,   7126672896,   5681332224,  3353143392,  1479513600,   488847744,  120185856,   21634704,   2765952,   237600,   12288,    289;
  9: 10077696, 362797056, 3401222400, 15479341056, 42578265600, 78726961152, 103891527936, 101475680256, 75042303552, 42592262400, 18683371872, 6342230016, 1658887776, 330728832, 49280400, 5308416, 390150, 17496, 361;
%F Formula
The row polynomials satisfy (36+(108*n-36)*x+(36*n^2+48*n-36)*x^2+(24*n^2-6*n-6)*x^3+(4*n^2-4*n+1)*x^4)*P(n,x) = (216+(648*n+648)*x+(216*n^2+1368*n+756)*x^2+(360*n^2+1080*n+432)*x^3+(204*n^2+396*n+126)*x^4+(48*n^2+66*n+18)*x^5+(4*n^2+4*n+1)*x^6)*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[((216 + (648*n + 648)*x + (216*n^2 + 1368*n + 756)*x^2 + (360*n^2 + 1080*n + 432)*x^3 + (204*n^2 + 396*n + 126)*x^4 + (48*n^2 + 66*n + 18)*x^5 + (4*n^2 + 4*n + 1)*x^6)*P[n - 1, x])/(36 + (108*n - 36)*x + (36*n^2 + 48*n - 36)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (4*n^2 - 4*n + 1)*x^4)]];
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] = ((216 + (648*n + 648)*x + (216*n^2 + 1368*n + 756)*x^2 + (360*n^2 + 1080*n + 432)*x^3 + (204*n^2 + 396*n + 126)*x^4 + (48*n^2 + 66*n + 18)*x^5 + (4*n^2 + 4*n + 1)*x^6)*v[i - 1])/(36 + (108*n - 36)*x + (36*n^2 + 48*n - 36)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (4*n^2 - 4*n + 1)*x^4)); 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

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.