Real-rooted atlas

A166344 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-8\,n x + 16\,n x^{2} - 4\,n^{2} x + 25\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 17\,x^{2} - 23\,x^{3} - 2\,n x + 16\,n x^{2} - 29\,n x^{3} + 4\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 9\,x^{3} - 7\,x^{4} - n x^{2} + 12\,n x^{3} - 11\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(1 + 4\,x + n + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 8\,x^{2} + 2\,n x - 12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 24\,x + 9\,x^{2}\)
\(P_{2}(x) = 1 + 60\,x + 135\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
1124934
216013516212
31136810544251516
412923654649618253612304
51608142475196839575547249111920
6112445094933451252267520379615239641128480
712520172440188150452267504233384938840403208112495840
81507656268096719364377285063032904296203044001280102789100150769920
9110192179082946690304324651050756394848636316058186761216161174612548001211968986880
1012042856018072154144802206884050781085167210631436998564962316810905358056224230061794714427673228800
111409041731101496124560014063686375722884142881488401179721285127365124556634385560077850002327369661472544169416491891200
12181860530374504183508896853378943756159958611481829021336940238892582784013859906875833413515775003163778602684859920034585851966683349657600
13116377616150675517865501184498593531025492484084588820349098657643382228132454403363830594426713680113460800241319269016116078951065630327379558025024225113908815820800
141327612489679209751946052962827945802325374467666067642094762465614215449882371227526897507376084834275025440029001258627433171791628896782139279424301843910664124996184281752562055024799027200
151655288148009453231296599260815667119431700273547503104760202699503707010870959221786005761241551327695269411087378404063400500543672419353210941900124190401059097273582923833833409448370023633004193843228939124745307648000
Technical details
Node id
n-81c4a92444b63754 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-bb1885632f63f3d0

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 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,    24,       9;
   2: 1,    60,     135,        16;
   3: 1,   136,     810,       544,         25;
   4: 1,   292,    3654,      6496,       1825,         36;
   5: 1,   608,   14247,     51968,      39575,       5472,          49;
   6: 1,  1244,   50949,    334512,     522675,     203796,       15239,         64;
   7: 1,  2520,  172440,   1881504,    5226750,    4233384,      938840,      40320,         81;
   8: 1,  5076,  562680,   9671936,   43772850,   63032904,    29620304,    4001280,     102789,      100;
   9: 1, 10192, 1790829,  46690304,  324651050,  756394848,   636316058,  186761216,   16117461,   254800,    121;
  10: 1, 20428, 5601807, 215414480, 2206884050, 7810851672, 10631436998, 5649623168, 1090535805, 62242300, 617947, 144;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -((4*n^2+8*n)*x-(9*n^3+25*n^2+16*n)*x^2)*P(n-2,x) - ((2*n+4)*x-(4*n^2+16*n+17)*x^2-(4*n^3-2*n^2-29*n-23)*x^3)*d/dx P(n-2,x) - ((n+2)*x^2-(4*n^2+12*n+9)*x^3+(4*n^2+11*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (n+1+(4*n^2+8*n+4)*x)*P(n-1,x) + ((2*n+3)*x-(4*n^2+12*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 24*x + 9*x^2, P(2,x) = 1 + 60*x + 135*x^2 + 16*x^3.
%t Mathematica
P[1, x_] := 1 + 24*x + 9*x^2;
P[2, x_] := 1 + 60*x + 135*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*P[n - 2, x] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*D[P[n - 2, x], x] - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 1 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 24*x + 9*x^2; v[2] = 1 + 60*x + 135*x^2 + 16*x^3; for(n = 3, N, my(i = n); v[i] = (-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*v[i - 2] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*deriv(v[i - 2], x) - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 1 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.