Real-rooted atlas

A132812 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-32\,x^{2} - 64\,x^{3} + 32\,n x^{2} + 64\,n x^{3}\right)\,P_{n-3}(x) + \left(4 + 16\,x - 8\,n - 32\,n x - 48\,n x^{2}\right)\,P_{n-2}(x) + \left(6 + 12\,x + 6\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 8\,x\)
\(P_{2}(x) = 15 + 60\,x + 48\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2156048123
3563365762561224
4210168044804480128012130
579279202880046080307206144120456
6300336036166320369600403200193536286721200367
711440160160896896256256039424003153920114688013107212005328
8437587001284612608161441283228825636900864227082246488064589824120475854
9167960302328022913280950476802348236803522355203130982401533542403538944026214401212674760
106466461293292011085360053209728015682867202927468544344408064024600576009840230401874329601153433612239414286
112496144549151685252755202865139200982333440022004269056324273438723088318464018166579200605552640096888422450331648123827279472
129657700231784800244764748814957845760585307008001532440166402724338073603269205688322580951859201274544128003598712832049073356802181038081255438195908
13374421609734961601124928896076149032960335055745024100516723507220977403166723051258642432305125864243220341724282888564936540162076348252162442762649693952409612752557899984
14145422675407183490051108548400379663502400185613267840062822952192001507750852608025847157473280314661047500802669851312128015256293212160554774298624011679458918401197893222404026531840129758499021015
Technical details
Node id
n-5a9de1acae37b77a · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-c585b23acdf03520

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:      4,       8;
  2:     15,      60,       48;
  3:     56,     336,      576,      256;
  4:    210,    1680,     4480,     4480,      1280;
  5:    792,    7920,    28800,    46080,     30720,      6144;
  6:   3003,   36036,   166320,   369600,    403200,    193536,     28672;
  7:  11440,  160160,   896896,  2562560,   3942400,   3153920,   1146880,    131072;
  8:  43758,  700128,  4612608, 16144128,  32288256,  36900864,  22708224,   6488064,   589824;
  9: 167960, 3023280, 22913280, 95047680, 234823680, 352235520, 313098240, 153354240, 35389440, 2621440;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = ((32*n-32)*x^2+(64*n-64)*x^3)*P(n-3,x) - (8*n-4+(32*n-16)*x+48*n*x^2)*P(n-2,x) + (6*n+6+(12*n+12)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 8*x, P(2,x) = 15 + 60*x + 48*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[2, x_] := 15 + 60*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((32*n - 32)*x^2 + (64*n - 64)*x^3)*P[n - 3, x] - (8*n - 4 + (32*n - 16)*x + 48*n*x^2)*P[n - 2, x] + (6*n + 6 + (12*n + 12)*x)*P[n - 1, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 8*x; v[3] = 15 + 60*x + 48*x^2; for(n = 3, N, my(i = n + 1); v[i] = (((32*n - 32)*x^2 + (64*n - 64)*x^3)*v[i - 3] - (8*n - 4 + (32*n - 16)*x + 48*n*x^2)*v[i - 2] + (6*n + 6 + (12*n + 12)*x)*v[i - 1])/(n + 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023