Real-rooted atlas

A143491 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 12\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 56 + 180\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
056180144380
15042292345617287980
2504028968620645875220736175560
355440379128103032013910409331202488324037880
4665280521481616913376290563202788992014183424298598496909120
5864864075775968282451680580692672711244800519063552209018880358318082422728000
6121080960116464723248636351361151911756816925739264158018273289155026944300987187242998169662990928000
7181621440018922680000869302938242311503851523921154997764401362810883269473320961550084014084256818790451597803521700755056000

All 15 rows

Technical details
Node id
n-979b7244b19f98cf · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-6f88f43a262894ce

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           56,           180,            144;
  1:          504,          2292,           3456,            1728;
  2:         5040,         28968,          62064,           58752,           20736;
  3:        55440,        379128,        1030320,         1391040,          933120,          248832;
  4:       665280,       5214816,       16913376,        29056320,        27889920,        14183424,         2985984;
  5:      8648640,      75775968,      282451680,       580692672,       711244800,       519063552,       209018880,        35831808;
  6:    121080960,    1164647232,     4863635136,     11519117568,     16925739264,     15801827328,      9155026944,      3009871872,       429981696;
  7:   1816214400,   18922680000,    86930293824,    231150385152,    392115499776,    440136281088,    326947332096,    155008401408,     42568187904,     5159780352;
  8:  29059430400,  324557452800,  1617956861184,   4741569688320,   9047652618240,  11747566494720,  10512792686592,   6403502407680,   2541191823360,   593374740480,   61917364224;
  9: 494010316800, 5866189862400, 31399956073728, 100022167035648, 210708930769920, 308280461829120, 319688273608704, 235013053169664, 120042289889280, 40581672468480, 8173092077568, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (n+8+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 56 + 180*x + 144*x^2.
%t Mathematica
P[0, x_] := 56 + 180*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 8 + 12*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] = 56 + 180*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 8 + 12*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.