Real-rooted atlas

A178820 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\,n + 3\,n^{2} x\right)\,P_{n}(x) = \left(12 + 36\,x + 27\,x^{2} + 4\,n + 30\,n x + 36\,n x^{2} + 6\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
182432
240240270550
31601440324021607000
4560672022680302401417574375
517922688012096024192022680081648700000
65376967685443201451520204120014696644286526037500
7153603225602177280725760013608000146966408573040209952048750000
84224010137607983360319334407484400010777536094303440461894409743085373828125
9112640304128027371520127733760359251200646652160754427520554273280233834040433026002750000000
1029286487859208895744047443968015567552003362591232490377888048037017603039842520112586760018576815419550781250
11745472246005762767564801660538880622702080015692092416274611617283362591232028371863520157621464005201508312773778096135078125000
121863680670924808302694405535129600233513280006725182464013730580864020175547392021278897640015762146400078022624680232133428803143473515910888671875
13458752017891328024153292801771241472083026944000269007298560627683696640107602919424013618494489601260971712000832241329920371413486080100591152480124994923206015625000000
1411141120467927040684343296054747463680282291609600101624979456026676557107205226427514880771714687744085746076416007074051304320420935284224017100495921604249827388804878628380039013671875000
Technical details
Node id
n-8b2e4bb20a839442 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      8,      24;
  2:     40,     240,      270;
  3:    160,    1440,     3240,      2160;
  4:    560,    6720,    22680,     30240,     14175;
  5:   1792,   26880,   120960,    241920,    226800,     81648;
  6:   5376,   96768,   544320,   1451520,   2041200,   1469664,    428652;
  7:  15360,  322560,  2177280,   7257600,  13608000,  14696640,   8573040,   2099520;
  8:  42240, 1013760,  7983360,  31933440,  74844000, 107775360,  94303440,  46189440,   9743085;
  9: 112640, 3041280, 27371520, 127733760, 359251200, 646652160, 754427520, 554273280, 233834040, 43302600;
%F Formula
The row polynomials satisfy (2*n+3*n^2*x)*P(n,x) = (4*n+12+(6*n^2+30*n+36)*x+(9*n^2+36*n+27)*x^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[((4*n + 12 + (6*n^2 + 30*n + 36)*x + (9*n^2 + 36*n + 27)*x^2)*P[n - 1, x])/(2*n + 3*n^2*x)]];
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] = ((4*n + 12 + (6*n^2 + 30*n + 36)*x + (9*n^2 + 36*n + 27)*x^2)*v[i - 1])/(2*n + 3*n^2*x)); 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.