Real-rooted atlas

A121574 → basis A013611 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(16\,n\right)\,P_{n}(x) = \left(224 + 128\,x - 160\,n - 64\,n x + 48\,n^{2}\right)\,P_{n-1}(x) + \left(816 + 960\,x + 256\,x^{2} - 276\,n - 288\,n x - 64\,n x^{2}\right)\,P_{n-1}'(x) + \left(345 + 636\,x + 368\,x^{2} + 64\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2546016130
3413684368641529
4316169565632198425617989
52419266408718243820899841024211640
61851496090488243845872002319364812840962489941
71417003543274088316327886272417510413096962252801638429294111
81084476647483028901104969676396864279296270366727028736103219265536344644690
98299837740859197288659470411125476488894315524696227841637130243632332846530562621444054738589
106352124693472931516848101540812178474176113846638087237386240316777267294239129618232115220709376104857647703926689
114861479168292263238567932617532812828249792013730389171210210619904053821009920201042984965214371840894173184912261124194304561235840880
1237206416521243935691120728485325824131009861926415798944494081345684546560829960273920373397176320121598967808279495311364303355904398458880167772166602929589881
132847523114872021953039876658867621731213043348829056174993050731521679911618560011869989949440624942686208024512705331207073709424641459544391682038851174417280532486710886477683348056611
14217929826301416661184646764588267150742881271225810722561878083975523842006390341222401598189339443209641277480960044258184724480153828222566403984479748096745743056896953281413127449083904268435456913943194931650
Technical details
Node id
n-7ed1ee07114ce3f2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bdd2ae59e2c72e1b

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:        7,         4;
  2:       54,        60,        16;
  3:      413,       684,       368,         64;
  4:     3161,      6956,      5632,       1984,       256;
  5:    24192,     66408,     71824,      38208,      9984,      1024;
  6:   185149,    609048,    824384,     587200,    231936,     48128,      4096;
  7:  1417003,   5432740,   8831632,    7886272,   4175104,   1309696,    225280,    16384;
  8: 10844766,  47483028,  90110496,   96763968,  64279296,  27036672,   7028736,  1032192,   65536;
  9: 82998377, 408591972, 886594704, 1112547648, 889431552, 469622784, 163713024, 36323328, 4653056, 262144;
%F Formula
The row polynomials satisfy 16*n*P(n,x) = (48*n^2-160*n+224-(64*n-128)*x)*P(n-1,x) - (276*n-816+(288*n-960)*x+(64*n-256)*x^2)*d/dx P(n-1,x) + (345+636*x+368*x^2+64*x^3)*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[((48*n^2 - 160*n + 224 - (64*n - 128)*x)*P[n - 1, x] - (276*n - 816 + (288*n - 960)*x + (64*n - 256)*x^2)*D[P[n - 1, x], x] + (345 + 636*x + 368*x^2 + 64*x^3)*D[P[n - 1, x], {x, 2}])/(16*n)]];
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] = ((48*n^2 - 160*n + 224 - (64*n - 128)*x)*v[i - 1] - (276*n - 816 + (288*n - 960)*x + (64*n - 256)*x^2)*deriv(v[i - 1], x) + (345 + 636*x + 368*x^2 + 64*x^3)*deriv(deriv(v[i - 1], x), x))/(16*n)); 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

Candidate equalities ?

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