Real-rooted atlas

A075502 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(-7\,x^{2}\right)\,P_{n-2}'(x) + \left(7 + 2\,x\right)\,P_{n-1}(x) + \left(7\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 49 + 42\,x + 3\,x^{2}\)
\(P_{1}(x) = 343 + 686\,x + 126\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
04942394
134368612641159
22401102903675280516651
31680714886292610127405256271550
4117649211768221681034802003430088274918823
58235432988284648706686163364041800750782041372897629813
657648014200069301067664675522361560834467555445468158466201692104850680
7403536075891626622230509685701604967658035735883753288457621411788029400028351048949474241
828247524982563479922492907779903480125569000145137688850180998280541075429509325987205093553850111220225362423
919773267431156453669406104749085978461410072630520456799760710009343580048887381251200430545906166876292583545050821232425336681176
10138412872011619430602517022170776105887540878688134284021642507036650546059007490496470689742723225541423107277684787181348847501310309655213914157083909733
1196889010407226747966926782468015444127413011741643025678020808586106450622521941883990406382843734826732801968250828544077675516919018073575520249267018197979682811427233542857394741
126782230728493174665314995762986233652171469033350062106406834002976821910398154001018589460371142661649418711653813114272187298969607107839748009021282115855003907449345843823052028983501029015854340708443961894
13474756150994344446670856086366207585266753257860695116717182286644041083763444969439325046352943628065776529270506057536683969877474843598762460835601123797502279508081851005349768740316794380838167384995804129720126001628137046911739497139
1433232930569601622262887108229010436595760244682802752690946116671127081203912067953870895942552076875245775095332845083332310520409199865908723189909279684943858790704103272271606780893450066267808200794212508910393616153261385914120001980057477001523217970212776488752849559
Technical details
Node id
n-5d7d7e644f711df6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1b31ab8c99ad484d

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:         49,            42,              3;
  1:        343,           686,            126,              4;
  2:       2401,         10290,           3675,            280,             5;
  3:      16807,        148862,          92610,          12740,           525,            6;
  4:     117649,       2117682,        2168103,         480200,         34300,          882,           7;
  5:     823543,      29882846,       48706686,       16336404,       1800750,        78204,        1372,          8;
  6:    5764801,     420006930,     1067664675,      522361560,      83446755,      5445468,      158466,       2016,        9;
  7:   40353607,    5891626622,    23050968570,    16049676580,    3573588375,    328845762,    14117880,     294000,     2835,     10;
  8:  282475249,   82563479922,   492907779903,   480125569000,  145137688850,  18099828054,  1075429509,   32598720,   509355,   3850,   11;
  9: 1977326743, 1156453669406, 10474908597846, 14100726305204, 5679976071000, 934358004888, 73812512004, 3054590616, 68762925, 835450, 5082, 12;
%F Formula
The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 7*x^2*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 49 + 42*x + 3*x^2, P(1,x) = 343 + 686*x + 126*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 49 + 42*x + 3*x^2;
P[1, x_] := 343 + 686*x + 126*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 7*x^2*D[P[n - 2, x], x] + (7 + 2*x)*P[n - 1, x] + 7*x*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] = 49 + 42*x + 3*x^2; v[2] = 343 + 686*x + 126*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 7*x^2*deriv(v[i - 2], x) + (7 + 2*x)*v[i - 1] + 7*x*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.