A159854 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 30\,x + 192\,x^{2} + 384\,x^{3} + 256\,x^{4} + 6\,n x + 80\,n x^{2} + 288\,n x^{3} + 256\,n x^{4} + 4\,n^{2} x^{2} + 32\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n}(x) = \left(2 + 76\,x + 696\,x^{2} + 2512\,x^{3} + 3968\,x^{4} + 2304\,x^{5} + 12\,n x + 200\,n x^{2} + 1056\,n x^{3} + 2176\,n x^{4} + 1536\,n x^{5} + 8\,n^{2} x^{2} + 80\,n^{2} x^{3} + 256\,n^{2} x^{4} + 256\,n^{2} x^{5}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 32\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	32	144															177
1	2	80	576	1024														1682
2	4	192	1872	6144	6400													14612
3	8	448	5472	25600	51200	36864												119592
4	16	1024	14976	90112	262400	368640	200704											937872
5	32	2304	39168	286720	1088000	2248704	2408448	1048576										7121952
6	64	5120	99072	851968	3968000	10764288	17059840	14680064	5308416									52736832
7	128	11264	244224	2408448	13260800	44384256	92725248	118489088	84934656	26214400								382672512
8	256	24576	589824	6553600	41574400	165150720	427098112	721420288	769720320	471859200	126877696							2730868992
9	512	53248	1400832	17301504	124108800	569769984	1753350144	3674210304	5191630848	4744806400	2537553920	603979776						19218166272
10	1024	114688	3280896	44564480	356352000	1854406656	6608781312	16508780544	28983951360	35127296000	28039970816	13287555072	2835349504					133650404352
11	2048	245760	7593984	112459776	991232000	5761400832	23313776640	67545071616	141543604224	213385216000	226096054272	160054640640	68048388096	13153337344				920013023232
12	4096	524288	17399808	278921216	2685337600	17232297984	77995180032	256892731392	625034133504	1125751193600	1484976553984	1396401242112	887464394752	341986770944	60397977600			6277114662912
13	8192	1114112	39518208	681574400	7113932800	49932140544	249810649088	921270484992	2550587719680	5338090700800	8418588884992	9861781782528	8330256842752	4800968130560	1691143372800	274877906944		42495144763392
14	16384	2359296	89063424	1644167168	18487705600	140840534016	771474063360	3147674157056	9765107269632	23271676313600	42673536761856	59798829662208	62956100386816	48246441377792	25427548569600	8246337208320	1241245548544	285707035164672

Technical details

Node id: n-f9c72d93728b9394 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-225246347a30c19a

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:   1,    32,     144;
  1:   2,    80,     576,     1024;
  2:   4,   192,    1872,     6144,      6400;
  3:   8,   448,    5472,    25600,     51200,     36864;
  4:  16,  1024,   14976,    90112,    262400,    368640,     200704;
  5:  32,  2304,   39168,   286720,   1088000,   2248704,    2408448,    1048576;
  6:  64,  5120,   99072,   851968,   3968000,  10764288,   17059840,   14680064,    5308416;
  7: 128, 11264,  244224,  2408448,  13260800,  44384256,   92725248,  118489088,   84934656,   26214400;
  8: 256, 24576,  589824,  6553600,  41574400, 165150720,  427098112,  721420288,  769720320,  471859200,  126877696;
  9: 512, 53248, 1400832, 17301504, 124108800, 569769984, 1753350144, 3674210304, 5191630848, 4744806400, 2537553920, 603979776;

%F Formula

The row polynomials satisfy (1+(6*n+30)*x+(4*n^2+80*n+192)*x^2+(32*n^2+288*n+384)*x^3+(64*n^2+256*n+256)*x^4)*P(n,x) = (2+(12*n+76)*x+(8*n^2+200*n+696)*x^2+(80*n^2+1056*n+2512)*x^3+(256*n^2+2176*n+3968)*x^4+(256*n^2+1536*n+2304)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 32*x + 144*x^2.

%t Mathematica

P[0, x_] := 1 + 32*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (12*n + 76)*x + (8*n^2 + 200*n + 696)*x^2 + (80*n^2 + 1056*n + 2512)*x^3 + (256*n^2 + 2176*n + 3968)*x^4 + (256*n^2 + 1536*n + 2304)*x^5)*P[n - 1, x])/(1 + (6*n + 30)*x + (4*n^2 + 80*n + 192)*x^2 + (32*n^2 + 288*n + 384)*x^3 + (64*n^2 + 256*n + 256)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 32*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (12*n + 76)*x + (8*n^2 + 200*n + 696)*x^2 + (80*n^2 + 1056*n + 2512)*x^3 + (256*n^2 + 2176*n + 3968)*x^4 + (256*n^2 + 1536*n + 2304)*x^5)*v[i - 1])/(1 + (6*n + 30)*x + (4*n^2 + 80*n + 192)*x^2 + (32*n^2 + 288*n + 384)*x^3 + (64*n^2 + 256*n + 256)*x^4)); 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 basis_map_A013611 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    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
    
    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

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.