Real-rooted atlas

A056939 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-12 - 84\,x + 12\,x^{2} + 12\,x^{3} - 19\,n - 13\,n x + 43\,n x^{2} + 7\,n x^{3} + 28\,n^{2} + 50\,n^{2} x + 22\,n^{2} x^{2} + n^{2} x^{3} + 32\,n^{3} + 24\,n^{3} x + 3\,n^{3} x^{2} + 10\,n^{4} + 3\,n^{4} x + n^{5}\right)\,P_{n}(x) = \left(-12 - 84\,x + 12\,x^{2} + 12\,x^{3} - 33\,n - 115\,n x + 11\,n x^{2} + 37\,n x^{3} + 10\,n x^{4} + 25\,n^{2} - 20\,n^{2} x + 62\,n^{2} x^{2} + 61\,n^{2} x^{3} + 10\,n^{2} x^{4} + 76\,n^{3} + 116\,n^{3} x + 105\,n^{3} x^{2} + 30\,n^{3} x^{3} + 53\,n^{4} + 73\,n^{4} x + 30\,n^{4} x^{2} + 11\,n^{5} + 10\,n^{5} x\right)\,P_{n-1}(x) + \left(-12 - 72\,x - 12\,x^{2} + 72\,x^{3} + 24\,x^{4} + 199\,n + 490\,n x + 365\,n x^{2} + 17\,n x^{3} - 72\,n x^{4} - 15\,n x^{5} + 133\,n^{2} + 127\,n^{2} x - 133\,n^{2} x^{2} - 172\,n^{2} x^{3} - 45\,n^{2} x^{4} - 12\,n^{3} - 105\,n^{3} x - 138\,n^{3} x^{2} - 45\,n^{3} x^{3} - 12\,n^{4} - 27\,n^{4} x - 15\,n^{4} x^{2}\right)\,P_{n-1}'(x) + \left(12\,x + 78\,x^{2} + 156\,x^{3} + 132\,x^{4} + 48\,x^{5} + 6\,x^{6} + 62\,n + 251\,n x + 400\,n x^{2} + 313\,n x^{3} + 120\,n x^{4} + 18\,n x^{5} + 56\,n^{2} + 176\,n^{2} x + 202\,n^{2} x^{2} + 100\,n^{2} x^{3} + 18\,n^{2} x^{4} + 12\,n^{3} + 30\,n^{3} x + 24\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2128121
35559161131
4276416188281909
51477289518954854516798
6829620112175056660109568153737
74839314016115342679219197962247981443341
8291010981808130051685797629636652136428413613784233
91794320691764010781850871437039032709648241250767698183133209232
101129776049030400880350008454960046947600152539802819250278340131702401298225341
11724136403495197207111438407930916405282904002151784805292936075477605819552161530812730718719
124713099442505218080570045634472513045445652658176278740809686855128816671072018786240115403234232388125423592085
1331087457851804827563545437626819649947245095816381242133819747819128700047193159832533484202433439454732186327525594811240133157514
1420746732688130643485728360673645969573464368204580209723093389744130776176168697705532622461801053411859513122109009742632939799767853358812296860845265
Technical details
Node id
n-33a26069c53180ea · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-294ef4e149f0b05c

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:       3,       1;
  2:      12,       8,        1;
  3:      55,      59,       16,       1;
  4:     276,     416,      188,      28,       1;
  5:    1477,    2895,     1895,     485,      45,      1;
  6:    8296,   20112,    17505,    6660,    1095,     68,      1;
  7:   48393,  140161,   153426,   79219,   19796,   2247,     98,    1;
  8:  291010,  981808,  1300516,  857976,  296366,  52136,   4284,  136,   1;
  9: 1794320, 6917640, 10781850, 8714370, 3903270, 964824, 125076, 7698, 183, 1;
%F Formula
The row polynomials satisfy (n^5+10*n^4+32*n^3+28*n^2-19*n-12+(3*n^4+24*n^3+50*n^2-13*n-84)*x+(3*n^3+22*n^2+43*n+12)*x^2+(n^2+7*n+12)*x^3)*P(n,x) = (11*n^5+53*n^4+76*n^3+25*n^2-33*n-12+(10*n^5+73*n^4+116*n^3-20*n^2-115*n-84)*x+(30*n^4+105*n^3+62*n^2+11*n+12)*x^2+(30*n^3+61*n^2+37*n+12)*x^3+(10*n^2+10*n)*x^4)*P(n-1,x) - (12*n^4+12*n^3-133*n^2-199*n+12+(27*n^4+105*n^3-127*n^2-490*n+72)*x+(15*n^4+138*n^3+133*n^2-365*n+12)*x^2+(45*n^3+172*n^2-17*n-72)*x^3+(45*n^2+72*n-24)*x^4+15*n*x^5)*d/dx P(n-1,x) + (12*n^3+56*n^2+62*n+(30*n^3+176*n^2+251*n+12)*x+(24*n^3+202*n^2+400*n+78)*x^2+(6*n^3+100*n^2+313*n+156)*x^3+(18*n^2+120*n+132)*x^4+(18*n+48)*x^5+6*x^6)*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[((11*n^5 + 53*n^4 + 76*n^3 + 25*n^2 - 33*n - 12 + (10*n^5 + 73*n^4 + 116*n^3 - 20*n^2 - 115*n - 84)*x + (30*n^4 + 105*n^3 + 62*n^2 + 11*n + 12)*x^2 + (30*n^3 + 61*n^2 + 37*n + 12)*x^3 + (10*n^2 + 10*n)*x^4)*P[n - 1, x] - (12*n^4 + 12*n^3 - 133*n^2 - 199*n + 12 + (27*n^4 + 105*n^3 - 127*n^2 - 490*n + 72)*x + (15*n^4 + 138*n^3 + 133*n^2 - 365*n + 12)*x^2 + (45*n^3 + 172*n^2 - 17*n - 72)*x^3 + (45*n^2 + 72*n - 24)*x^4 + 15*n*x^5)*D[P[n - 1, x], x] + (12*n^3 + 56*n^2 + 62*n + (30*n^3 + 176*n^2 + 251*n + 12)*x + (24*n^3 + 202*n^2 + 400*n + 78)*x^2 + (6*n^3 + 100*n^2 + 313*n + 156)*x^3 + (18*n^2 + 120*n + 132)*x^4 + (18*n + 48)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 + 10*n^4 + 32*n^3 + 28*n^2 - 19*n - 12 + (3*n^4 + 24*n^3 + 50*n^2 - 13*n - 84)*x + (3*n^3 + 22*n^2 + 43*n + 12)*x^2 + (n^2 + 7*n + 12)*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] = ((11*n^5 + 53*n^4 + 76*n^3 + 25*n^2 - 33*n - 12 + (10*n^5 + 73*n^4 + 116*n^3 - 20*n^2 - 115*n - 84)*x + (30*n^4 + 105*n^3 + 62*n^2 + 11*n + 12)*x^2 + (30*n^3 + 61*n^2 + 37*n + 12)*x^3 + (10*n^2 + 10*n)*x^4)*v[i - 1] - (12*n^4 + 12*n^3 - 133*n^2 - 199*n + 12 + (27*n^4 + 105*n^3 - 127*n^2 - 490*n + 72)*x + (15*n^4 + 138*n^3 + 133*n^2 - 365*n + 12)*x^2 + (45*n^3 + 172*n^2 - 17*n - 72)*x^3 + (45*n^2 + 72*n - 24)*x^4 + 15*n*x^5)*deriv(v[i - 1], x) + (12*n^3 + 56*n^2 + 62*n + (30*n^3 + 176*n^2 + 251*n + 12)*x + (24*n^3 + 202*n^2 + 400*n + 78)*x^2 + (6*n^3 + 100*n^2 + 313*n + 156)*x^3 + (18*n^2 + 120*n + 132)*x^4 + (18*n + 48)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 + 10*n^4 + 32*n^3 + 28*n^2 - 19*n - 12 + (3*n^4 + 24*n^3 + 50*n^2 - 13*n - 84)*x + (3*n^3 + 22*n^2 + 43*n + 12)*x^2 + (n^2 + 7*n + 12)*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. [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