Real-rooted atlas

A143491 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 8\,x + 9\,x^{2} - 4\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(8\,x^{2} + 4\,x^{3} + 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(5 + 4\,x + 2\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 6 + 20\,x + 9\,x^{2}\)
\(P_{1}(x) = 24 + 104\,x + 81\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0620935
1241048116225
2120616639224251624
37204176522024805003613132
45040321124593626640737597249118124
540320277056439740294784100625183961715641172700
636288026547844581036343481613662253104644047428168112753576
73628800279993605178362442492224190291505072004827316810244374100150917976
8399168003225081606326184245594742402757147508319402016004037197184015066065001211931559552
947900160040297651208317064448783834585641827555001395357480305284749445653124303530264000931714426596717056
1062270208005430295296011718880934411668438848066623236900241628151605867938307978088320112348863874500044056112960169392156797824
118717829120078514942464017628249749761841917099776111504467360043421687337611503941248821357482432281077711226113230016749304705744175761966165817614720
121307674368000121259545344002820896083008030762667563264196036655724008118917430624231660859830447061779564869192220383738707970056720964330519216109190923324225102992244837120
132092278988800019924597002240047862677098368054235194471014436172531722600015813195731424048116486297880105556592261121702701173745203615769500180137207251163333808532882351639540303752561821602444624640
1435568742809600034708726499328008584958539272960100708750973767686996755306451600320912773114752010332154311861202422922297635844230542616171355635682960005526083334204121449113622712014788967392023985003891228934012249593822720
Technical details
Node id
n-c391da7eb58bddf2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-505ab518d44a108d

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:         6,         20,          9;
  1:        24,        104,         81,         16;
  2:       120,        616,        639,        224,         25;
  3:       720,       4176,       5220,       2480,        500,         36;
  4:      5040,      32112,      45936,      26640,       7375,        972,        49;
  5:     40320,     277056,     439740,     294784,     100625,      18396,      1715,       64;
  6:    362880,    2654784,    4581036,    3434816,    1366225,     310464,     40474,     2816,      81;
  7:   3628800,   27999360,   51783624,   42492224,   19029150,    5072004,    827316,    81024,    4374,    100;
  8:  39916800,  322508160,  632618424,  559474240,  275714750,   83194020,  16004037,  1971840,  150660,   6500,  121;
  9: 479001600, 4029765120, 8317064448, 7838345856, 4182755500, 1395357480, 305284749, 44565312, 4303530, 264000, 9317, 144;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+4*n+4+(4*n+8)*x-9*x^2)*P(n-2,x) + ((4*n+8)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n+5+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 6 + 20*x + 9*x^2, P(1,x) = 24 + 104*x + 81*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 6 + 20*x + 9*x^2;
P[1, x_] := 24 + 104*x + 81*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n + 4 + (4*n + 8)*x - 9*x^2)*P[n - 2, x] + ((4*n + 8)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n + 5 + 4*x)*P[n - 1, 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] = 6 + 20*x + 9*x^2; v[2] = 24 + 104*x + 81*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n + 4 + (4*n + 8)*x - 9*x^2)*v[i - 2] + ((4*n + 8)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n + 5 + 4*x)*v[i - 1] - 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

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. [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.