Real-rooted atlas

A139548 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 8\,x + x^{2} + 12\,n x - 4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(32 + 64\,x + 48\,x^{2} + 16\,x^{3} + 2\,x^{4} + 96\,n x + 128\,n x^{2} + 56\,n x^{3} + 8\,n x^{4} + 32\,n^{2} x^{2} + 32\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18321858
2645128645121002052
35126144172802048012000345639260264
4409665536258048458752448000258048878081638412961597968
532768655360331776078643201075200092897285268480196608046656064000387239680928
62621446291456389283841153433602027520002335703041854504961038090244105728011264000204441622118410816941004864
72097152587202564293918721526726656328007680047233105924821712896359871283219926466568200192002480558085367398478740487024642880021563750016

All 15 rows

Technical details
Node id
n-b51d0aea8d328ed6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-86a3c7a2e1168367

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:         8,         32,          18;
  2:        64,        512,         864,          512,          100;
  3:       512,       6144,       17280,        20480,        12000,          3456,           392;
  4:      4096,      65536,      258048,       458752,       448000,        258048,         87808,         16384,          1296;
  5:     32768,     655360,     3317760,      7864320,     10752000,       9289728,       5268480,       1966080,        466560,         64000,         3872;
  6:    262144,    6291456,    38928384,    115343360,    202752000,     233570304,     185450496,     103809024,      41057280,      11264000,      2044416,       221184,        10816;
  7:   2097152,   58720256,   429391872,   1526726656,   3280076800,    4723310592,    4821712896,    3598712832,    1992646656,     820019200,    248055808,     53673984,      7874048,      702464,      28800;
  8:  16777216,  536870912,  4529848320,  18790481920,  47710208000,   82443239424,  102863208448,   95965675520,   68319313920,   37486592000,  15875571712,   5152702464,   1259847680,   224788480,   27648000,   2097152,    73984;
  9: 134217728, 4831838208, 46204452864, 219043332096, 641728512000, 1293724680192, 1907644956672, 2135672487936, 1858285338624, 1274544128000, 693989277696, 300328943616, 102803570688, 27514109952, 5640192000, 855638016, 90556416, 5971968, 184832;
%F Formula
The row polynomials satisfy (4+(12*n-8)*x+(4*n^2-4*n+1)*x^2)*P(n,x) = (32+(96*n+64)*x+(32*n^2+128*n+48)*x^2+(32*n^2+56*n+16)*x^3+(8*n^2+8*n+2)*x^4)*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[((32 + (96*n + 64)*x + (32*n^2 + 128*n + 48)*x^2 + (32*n^2 + 56*n + 16)*x^3 + (8*n^2 + 8*n + 2)*x^4)*P[n - 1, x])/(4 + (12*n - 8)*x + (4*n^2 - 4*n + 1)*x^2)]];
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] = ((32 + (96*n + 64)*x + (32*n^2 + 128*n + 48)*x^2 + (32*n^2 + 56*n + 16)*x^3 + (8*n^2 + 8*n + 2)*x^4)*v[i - 1])/(4 + (12*n - 8)*x + (4*n^2 - 4*n + 1)*x^2)); 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.