Real-rooted atlas

A136124 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 8\,x - 16\,x^{2} - 2\,n - 8\,n x - n^{2}\right)\,P_{n-2}(x) + \left(3 + 8\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2406016116
3248572384641268
4172851765360217625614696
513536486166584041280115201024181816
61183684871367963846563202828805836840962403552
71145952525052898938569894976562688017920002867201638433907296
812185280609471361285127041488596481031600644346675210723328137625665536509296704
91412956807606431361758580800228913971218446507529504122883108618246144000064880642621448123774400
101775381760101782494722542415385636422570240331018649601989737164879799623682094366720340131840301465601048576137245248000
11240379660801455622940163885117350406033273213446054027724804116771338241931599380486230217523213451919360183107584013841203241943042449406937600
123489855667202217721803264627068914329610435553193472113835444828168570493904896456116526694417252162273284586434068488307605504096311705606291456001677721646065366144000
135409140083200358833550003201067558751267841887020897448962211594243921921814656132157441071841127751684608861918003214397409787904321601221427249647242444849673142272283534950467108864910810699545600
14891615900672006147914519347201913883526232064356784606852710444516639559741443933354087190528253757423638118412135787111219204318292987740161135428761026562164430838169628851316981762519098982401268357529626843545618892020104294400
Technical details
Node id
n-e194b17d2b932ec0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a1428adee7f386f3

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:        40,        60,         16;
  3:       248,       572,        384,         64;
  4:      1728,      5176,       5360,       2176,        256;
  5:     13536,     48616,      65840,      41280,      11520,      1024;
  6:    118368,    487136,     796384,     656320,     282880,     58368,      4096;
  7:   1145952,   5250528,    9893856,    9894976,    5626880,   1792000,    286720,    16384;
  8:  12185280,  60947136,  128512704,  148859648,  103160064,  43466752,  10723328,  1376256,   65536;
  9: 141295680, 760643136, 1758580800, 2289139712, 1844650752, 950412288, 310861824, 61440000, 6488064, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(8*n+8)*x+16*x^2)*P(n-2,x) + (2*n+3+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n + 1 + (8*n + 8)*x + 16*x^2)*P[n - 2, x] + (2*n + 3 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n + 1 + (8*n + 8)*x + 16*x^2)*v[i - 2] + (2*n + 3 + 8*x)*v[i - 1]); 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. [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.