Real-rooted atlas

A038320 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 330\,x + 144\,x^{2} + 198\,n x + 144\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 6534\,x + 6732\,x^{2} + 1944\,x^{3} + 2178\,n x + 3564\,n x^{2} + 1296\,n x^{3} + 396\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 528\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0121528324973
11331871210692345624191
21464112777623522415206432400562105
3161051175692043124404181760178200027993612474107
4177156123191344711552609199872058806000184757762286144267684805
519487171297622248109579100417709753601509354000711317376176033088179159045598496151
621435888137415368321607160139231169166336332057880002086530969677454558721576599552136048896114725865457
7235794769146301518296227298362544514291244544657474602400516416414976255600043776780416778241346884070410077696002312258422355

All 15 rows

Technical details
Node id
n-6f9e931d97a0ef8a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-331d650033ed3318

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          121,           528,            324;
  1:         1331,          8712,          10692,            3456;
  2:        14641,        127776,         235224,          152064,           32400;
  3:       161051,       1756920,        4312440,         4181760,         1782000,          279936;
  4:      1771561,      23191344,       71155260,        91998720,        58806000,        18475776,         2286144;
  5:     19487171,     297622248,     1095791004,      1770975360,      1509354000,       711317376,       176033088,       17915904;
  6:    214358881,    3741536832,    16071601392,     31169166336,     33205788000,     20865309696,      7745455872,     1576599552,      136048896;
  7:   2357947691,   46301518296,   227298362544,    514291244544,    657474602400,    516416414976,    255600043776,    78041677824,    13468840704,    1007769600;
  8:  25937424601,  565907445840,  3125352484980,   8081719557120,  12053701044000,  11361161129472,   7029001203840,  2861528186880,   740786238720,  110854656000,   7316407296;
  9: 285311670611, 6847480094664, 42018627853620, 122236008301440, 208356832332000, 229116749444352, 170101829132928, 86561227653120, 29878378295040, 6706706688000, 885285282816, 52242776064;
%F Formula
The row polynomials satisfy (121+(198*n+330)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (1331+(2178*n+6534)*x+(396*n^2+3564*n+6732)*x^2+(216*n^2+1296*n+1944)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 528*x + 324*x^2.
%t Mathematica
P[0, x_] := 121 + 528*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1331 + (2178*n + 6534)*x + (396*n^2 + 3564*n + 6732)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*P[n - 1, x])/(121 + (198*n + 330)*x + (36*n^2 + 144*n + 144)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 528*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (2178*n + 6534)*x + (396*n^2 + 3564*n + 6732)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*v[i - 1])/(121 + (198*n + 330)*x + (36*n^2 + 144*n + 144)*x^2)); 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.