A038284 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(16 + 60\,x + 36\,x^{2} + 36\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(128 + 864\,x + 1224\,x^{2} + 486\,x^{3} + 288\,n x + 648\,n x^{2} + 324\,n x^{3} + 72\,n^{2} x^{2} + 54\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 64 + 384\,x + 324\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	64	384	324								772
1	512	4608	7776	3456							16352
2	4096	49152	124416	110592	32400						320656
3	32768	491520	1658880	2211840	1296000	279936					5970944
4	262144	4718592	19906560	35389440	31104000	13436928	2286144				107103808
5	2097152	44040192	222953472	495452160	580608000	376233984	128024064	17915904			1867324928
6	16777216	402653184	2378170368	6341787648	9289728000	8026324992	4096770048	1146617856	136048896		31834878208
7	134217728	3623878656	24461180928	76101451776	133772083200	144473849856	98322481152	41278242816	9795520512	1007769600	532970676224

All 15 rows

Technical details

Node id: n-30e58548258a254c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e997b32cda4979c2

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:         64,          384,           324;
  1:        512,         4608,          7776,          3456;
  2:       4096,        49152,        124416,        110592,          32400;
  3:      32768,       491520,       1658880,       2211840,        1296000,         279936;
  4:     262144,      4718592,      19906560,      35389440,       31104000,       13436928,        2286144;
  5:    2097152,     44040192,     222953472,     495452160,      580608000,      376233984,      128024064,       17915904;
  6:   16777216,    402653184,    2378170368,    6341787648,     9289728000,     8026324992,     4096770048,     1146617856,      136048896;
  7:  134217728,   3623878656,   24461180928,   76101451776,   133772083200,   144473849856,    98322481152,    41278242816,     9795520512,    1007769600;
  8: 1073741824,  32212254720,  244611809280,  869730877440,  1783627776000,  2311581597696,  1966449623040,  1100753141760,   391820820480,   80621568000,   7316407296;
  9: 8589934592, 283467841536, 2391759912960, 9567039651840, 22422749184000, 33903196766208, 34609513365504, 24216569118720, 11493410734080, 3547348992000, 643843842048, 52242776064;

%F Formula

The row polynomials satisfy (16+(36*n+60)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (128+(288*n+864)*x+(72*n^2+648*n+1224)*x^2+(54*n^2+324*n+486)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 384*x + 324*x^2.

%t Mathematica

P[0, x_] := 64 + 384*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((128 + (288*n + 864)*x + (72*n^2 + 648*n + 1224)*x^2 + (54*n^2 + 324*n + 486)*x^3)*P[n - 1, x])/(16 + (36*n + 60)*x + (9*n^2 + 36*n + 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 64 + 384*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((128 + (288*n + 864)*x + (72*n^2 + 648*n + 1224)*x^2 + (54*n^2 + 324*n + 486)*x^3)*v[i - 1])/(16 + (36*n + 60)*x + (9*n^2 + 36*n + 36)*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.