Real-rooted atlas

A081578 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(37 + 263\,x + 760\,x^{2} + 1150\,x^{3} + 965\,x^{4} + 427\,x^{5} + 78\,x^{6} - 51\,n - 365\,n x - 1069\,n x^{2} - 1644\,n x^{3} - 1403\,n x^{4} - 631\,n x^{5} - 117\,n x^{6} + 16\,n^{2} + 112\,n^{2} x + 327\,n^{2} x^{2} + 508\,n^{2} x^{3} + 442\,n^{2} x^{4} + 204\,n^{2} x^{5} + 39\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(36 + 339\,x + 1311\,x^{2} + 2730\,x^{3} + 3330\,x^{4} + 2391\,x^{5} + 939\,x^{6} + 156\,x^{7} - 16\,n - 152\,n x - 596\,n x^{2} - 1262\,n x^{3} - 1568\,n x^{4} - 1148\,n x^{5} - 460\,n x^{6} - 78\,n x^{7}\right)\,P_{n-1}'(x) + \left(4 + 48\,x + 241\,x^{2} + 664\,x^{3} + 1105\,x^{4} + 1144\,x^{5} + 723\,x^{6} + 256\,x^{7} + 39\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
27282560
320120228136504
461488141017367694464
5182182070761333612166430038880
6547656432067814961135058211624121340416
71640229601350964325608132088971685378001351842975616
84921787365419242093872496372673890406743572345006475782526023680
9147622657162094360948091227149148509760246274159248808080217783144247956227556864
1044287885740786647740832592137116902311125320482960010506356944343155339135753644238122491989909504
111328602922920288804441690975926516318001734764304325470069643026614883927444684235748500083761937213348060017400821760
1239858195659441040794626785841202951794071902232792019864894596317388445923651792475529507086232158940572705124968472748232641152162758656
13119574231089292369334812265365047212861915122443975849641122650608082105145391202926978060502980116160842159661397401054954052883113668071841942559641330596274176
14358722710044235612936437351015543959254273538943208838600108596549226939128488554168020966577293132579200413452235440727320515465227382806910458176051880387427402351111762511635483852800
Technical details
Node id
n-73af5bea60a816d7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4ff87de320dd4d46

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:     2,      4;
  2:     7,     28,      25;
  3:    20,    120,     228,     136;
  4:    61,    488,    1410,    1736,      769;
  5:   182,   1820,    7076,   13336,    12166,     4300;
  6:   547,   6564,   32067,   81496,   113505,    82116,    24121;
  7:  1640,  22960,  135096,  432560,   813208,   897168,   537800,   135184;
  8:  4921,  78736,  541924, 2093872,  4963726,  7389040,  6743572,  3450064,   757825;
  9: 14762, 265716, 2094360, 9480912, 27149148, 50976024, 62741592, 48808080, 21778314, 4247956;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (16*n^2-51*n+37+(112*n^2-365*n+263)*x+(327*n^2-1069*n+760)*x^2+(508*n^2-1644*n+1150)*x^3+(442*n^2-1403*n+965)*x^4+(204*n^2-631*n+427)*x^5+(39*n^2-117*n+78)*x^6)*P(n-1,x) - (16*n-36+(152*n-339)*x+(596*n-1311)*x^2+(1262*n-2730)*x^3+(1568*n-3330)*x^4+(1148*n-2391)*x^5+(460*n-939)*x^6+(78*n-156)*x^7)*d/dx P(n-1,x) + (4+48*x+241*x^2+664*x^3+1105*x^4+1144*x^5+723*x^6+256*x^7+39*x^8)*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[((16*n^2 - 51*n + 37 + (112*n^2 - 365*n + 263)*x + (327*n^2 - 1069*n + 760)*x^2 + (508*n^2 - 1644*n + 1150)*x^3 + (442*n^2 - 1403*n + 965)*x^4 + (204*n^2 - 631*n + 427)*x^5 + (39*n^2 - 117*n + 78)*x^6)*P[n - 1, x] - (16*n - 36 + (152*n - 339)*x + (596*n - 1311)*x^2 + (1262*n - 2730)*x^3 + (1568*n - 3330)*x^4 + (1148*n - 2391)*x^5 + (460*n - 939)*x^6 + (78*n - 156)*x^7)*D[P[n - 1, x], x] + (4 + 48*x + 241*x^2 + 664*x^3 + 1105*x^4 + 1144*x^5 + 723*x^6 + 256*x^7 + 39*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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] = ((16*n^2 - 51*n + 37 + (112*n^2 - 365*n + 263)*x + (327*n^2 - 1069*n + 760)*x^2 + (508*n^2 - 1644*n + 1150)*x^3 + (442*n^2 - 1403*n + 965)*x^4 + (204*n^2 - 631*n + 427)*x^5 + (39*n^2 - 117*n + 78)*x^6)*v[i - 1] - (16*n - 36 + (152*n - 339)*x + (596*n - 1311)*x^2 + (1262*n - 2730)*x^3 + (1568*n - 3330)*x^4 + (1148*n - 2391)*x^5 + (460*n - 939)*x^6 + (78*n - 156)*x^7)*deriv(v[i - 1], x) + (4 + 48*x + 241*x^2 + 664*x^3 + 1105*x^4 + 1144*x^5 + 723*x^6 + 256*x^7 + 39*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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. [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