Real-rooted atlas

A075500 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-20\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-50\,x + 20\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-25\,x^{2} + 20\,x^{3}\right)\,P_{n-2}''(x) + \left(5 + 4\,x\right)\,P_{n-1}(x) + \left(10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 25 + 60\,x + 9\,x^{2}\)
\(P_{1}(x) = 125 + 700\,x + 270\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02560994
1125700270161111
2625750056258002514575
331257750010125026000187536209786
4156257875001693125700000875003780493287579
57812579375002716875017010000328125023940068606455721949
6390625796875004253906253885000001086093751190700056595011520811015062676
719531257984375006560156250852625000033222656255136075003601500012000001822510019759903325
8976562579921875001001988281251821875000009637890625020192287500195960187595040000233887527500121409016483371
948828125799609375001520964843750382188125000026941406250007445542500009607001250063610800002255343754262500399301448964211663824
1024414062579980468750022994384765625791415625000007332520507812526216190000000437587150000037992240000018199788750491562500735377556160169207251883833229
1112207031257999023437500346715332031250162371015625000019567888183593758920739953125001888386500000002091232000000012998297812504704700000099924825012121200768951965038387404321541
12610351562579995117187500521872778320312533090585937500000514567675781250002957999575781250078235645769531251083139200000000849594951562503957078125000111885523750191646000019223750102900225128421812448888750
1330517578125799975585937500784609057617187506710894570312500001338123229980468750961497618046875000314086421085937500535441013250000005194025332031250302742171875000109417683375002481406200003498722500294980001350002563423109700275985131
1415258789062579998779296875001178713531494140625135612751953125000003450165802612304687530771825992578125000123017298292382812502551999378500000000301497893180859375215494855468750009681152865312502791004580000051863755125061225500004398750017408028995199245912628000619
Technical details
Node id
n-5d54cd7ef0177801 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1cc6c3f42efa30d8

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 2)
  • 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:       25,          60,             9;
  1:      125,         700,           270,            16;
  2:      625,        7500,          5625,           800,            25;
  3:     3125,       77500,        101250,         26000,          1875,           36;
  4:    15625,      787500,       1693125,        700000,         87500,         3780,          49;
  5:    78125,     7937500,      27168750,      17010000,       3281250,       239400,        6860,         64;
  6:   390625,    79687500,     425390625,     388500000,     108609375,     11907000,      565950,      11520,        81;
  7:  1953125,   798437500,    6560156250,    8526250000,    3322265625,    513607500,    36015000,    1200000,     18225,     100;
  8:  9765625,  7992187500,  100198828125,  182187500000,   96378906250,  20192287500,  1959601875,   95040000,   2338875,   27500,   121;
  9: 48828125, 79960937500, 1520964843750, 3821881250000, 2694140625000, 744554250000, 96070012500, 6361080000, 225534375, 4262500, 39930, 144;
%F Formula
The row polynomials satisfy P(n,x) = -(20*x-9*x^2)*P(n-2,x) - (50*x-20*x^2-4*x^3)*d/dx P(n-2,x) - (25*x^2-20*x^3)*d^2/dx^2 P(n-2,x) + (5+4*x)*P(n-1,x) + (10*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 25 + 60*x + 9*x^2, P(1,x) = 125 + 700*x + 270*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 25 + 60*x + 9*x^2;
P[1, x_] := 125 + 700*x + 270*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(20*x - 9*x^2)*P[n - 2, x] - (50*x - 20*x^2 - 4*x^3)*D[P[n - 2, x], x] - (25*x^2 - 20*x^3)*D[P[n - 2, x], {x, 2}] + (5 + 4*x)*P[n - 1, x] + (10*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 25 + 60*x + 9*x^2; v[2] = 125 + 700*x + 270*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(20*x - 9*x^2)*v[i - 2] - (50*x - 20*x^2 - 4*x^3)*deriv(v[i - 2], x) - (25*x^2 - 20*x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 4*x)*v[i - 1] + (10*x - 4*x^2)*deriv(v[i - 1], x)); 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.