A013618 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 13\,x + 12\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 24\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	24	144															169
1	1	36	432	1728														2197
2	1	48	864	6912	20736													28561
3	1	60	1440	17280	103680	248832												371293
4	1	72	2160	34560	311040	1492992	2985984											4826809
5	1	84	3024	60480	725760	5225472	20901888	35831808										62748517
6	1	96	4032	96768	1451520	13934592	83607552	286654464	429981696									815730721
7	1	108	5184	145152	2612736	31352832	250822656	1289945088	3869835264	5159780352								10604499373
8	1	120	6480	207360	4354560	62705664	627056640	4299816960	19349176320	51597803520	61917364224							137858491849
9	1	132	7920	285120	6842880	114960384	1379524608	11824496640	70946979840	283787919360	681091006464	743008370688						1792160394037
10	1	144	9504	380160	10264320	197074944	2759049216	28378791936	212840939520	1135151677440	4086546038784	8916100448256	8916100448256					23298085122481
11	1	156	11232	494208	14826240	320246784	5123948544	61487382528	553386442752	3689242951680	17708366168064	57954652913664	115909305827328	106993205379072				302875106592253
12	1	168	13104	628992	20756736	498161664	8966909952	122974765056	1291235033088	10329880264704	61979281588224	270455046930432	811365140791296	1497904875307008	1283918464548864			3937376385699289
13	1	180	15120	786240	28304640	747242496	14944849920	230577684480	2766932213760	25824700661760	185937844764672	1014206425989120	4056825703956480	11234286564802560	19258776968232960	15407021574586368		51185893014090757
14	1	192	17280	967680	37739520	1086898176	23911759872	409915883520	5533864427520	59027887226880	495834252705792	3245460563165184	16227302815825920	59916195012280320	154070215745863680	246512345193381888	184884258895036416	665416609183179841

Technical details

Node id: n-02a5702f23ec3d86 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-a8581dd4789601b6

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: 1,  24,  144;
  1: 1,  36,  432,   1728;
  2: 1,  48,  864,   6912,   20736;
  3: 1,  60, 1440,  17280,  103680,    248832;
  4: 1,  72, 2160,  34560,  311040,   1492992,    2985984;
  5: 1,  84, 3024,  60480,  725760,   5225472,   20901888,    35831808;
  6: 1,  96, 4032,  96768, 1451520,  13934592,   83607552,   286654464,   429981696;
  7: 1, 108, 5184, 145152, 2612736,  31352832,  250822656,  1289945088,  3869835264,   5159780352;
  8: 1, 120, 6480, 207360, 4354560,  62705664,  627056640,  4299816960, 19349176320,  51597803520,  61917364224;
  9: 1, 132, 7920, 285120, 6842880, 114960384, 1379524608, 11824496640, 70946979840, 283787919360, 681091006464, 743008370688;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (1+13*x+12*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 24*x + 144*x^2.

%t Mathematica

P[0, x_] := 1 + 24*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 13*x + 12*x^2)*P[n - 1, x])/(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 + 24*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 13*x + 12*x^2)*v[i - 1])/(1 + 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of coefficients in expansion of (1+11x)^n.?
  - 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.