Real-rooted atlas

A036969 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 7\,x + 16\,x^{2} + 17\,x^{3} + 9\,x^{4} + 2\,x^{5} - 3\,n x - 12\,n x^{2} - 18\,n x^{3} - 12\,n x^{4} - 3\,n x^{5} + n^{2} x^{2} + 3\,n^{2} x^{3} + 3\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(3\,x + 16\,x^{2} + 34\,x^{3} + 36\,x^{4} + 19\,x^{5} + 4\,x^{6} - 2\,n x^{2} - 8\,n x^{3} - 12\,n x^{4} - 8\,n x^{5} - 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(x^{2} + 5\,x^{3} + 10\,x^{4} + 10\,x^{5} + 5\,x^{6} + x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
217715
31245937121
41894085832641345
51346278269076902243319371
61137119853771621280659729827913351663
7154681512698704132139777263282216045613869067791217
8121853122474210144925348515836193862160255508305725536346119205355905
9187390103671201232800835734525971385580997191629049415359980556659956851211593736332211795
101349535902454751569561960970727585430727487627567420808466393114005043453232888164194706592655174768225296019663
1111398112798227935209711370451705072127106899914658521636451979474243265348714523056776965601357892980874454104255980660289036339135078745321
1215592417712113645229328803675531165776617701586575914106247103128123964888202362633541101568180644158993168481076046664580166326139828304095721845579100993417787608507841
13122369634638106073144265647984027592870450868403759408239983771372688537926608319747614704317849864717344372995293887876063400379018659207884617558114541788044761910987626735750684726216221369777637799163
14189478499573065136073639766676681981172892527170079921293453940679240856861251730283115364111519130336220170901303356441291999291064463833270547575863116081172286246394107794720232830496716191782883893833647519833121524113511213510798425183855
Technical details
Node id
n-b96fe6111cec1c2e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4c47ff87ed3f3de6

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: 1,     2;
  2: 1,     7,        7;
  3: 1,    24,       59,        37;
  4: 1,    89,      408,       583,       264;
  5: 1,   346,     2782,      6907,      6902,       2433;
  6: 1,  1371,    19853,     77162,    128065,      97298,      27913;
  7: 1,  5468,   151269,    870413,   2139777,    2632822,    1604561,     386906;
  8: 1, 21853,  1224742,  10144925,  34851583,   61938621,   60255508,   30572553,   6346119;
  9: 1, 87390, 10367120, 123280083, 573452597, 1385580997, 1916290494, 1535998055, 665995685, 121159373;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(3*n-7)*x+(n^2-12*n+16)*x^2+(3*n^2-18*n+17)*x^3+(3*n^2-12*n+9)*x^4+(n^2-3*n+2)*x^5)*P(n-1,x) + (3*x-(2*n-16)*x^2-(8*n-34)*x^3-(12*n-36)*x^4-(8*n-19)*x^5-(2*n-4)*x^6)*d/dx P(n-1,x) + (x^2+5*x^3+10*x^4+10*x^5+5*x^6+x^7)*d^2/dx^2 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[((1 - (3*n - 7)*x + (n^2 - 12*n + 16)*x^2 + (3*n^2 - 18*n + 17)*x^3 + (3*n^2 - 12*n + 9)*x^4 + (n^2 - 3*n + 2)*x^5)*P[n - 1, x] + (3*x - (2*n - 16)*x^2 - (8*n - 34)*x^3 - (12*n - 36)*x^4 - (8*n - 19)*x^5 - (2*n - 4)*x^6)*D[P[n - 1, x], x] + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*D[P[n - 1, x], {x, 2}])/(1 + 2*x + 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] = ((1 - (3*n - 7)*x + (n^2 - 12*n + 16)*x^2 + (3*n^2 - 18*n + 17)*x^3 + (3*n^2 - 12*n + 9)*x^4 + (n^2 - 3*n + 2)*x^5)*v[i - 1] + (3*x - (2*n - 16)*x^2 - (8*n - 34)*x^3 - (12*n - 36)*x^4 - (8*n - 19)*x^5 - (2*n - 4)*x^6)*deriv(v[i - 1], x) + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*deriv(deriv(v[i - 1], x), x))/(1 + 2*x + 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.