basis-map coefficients of A371080, shifted → 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

\(P_{n}(x) = \left(-25 + 80\,x + 144\,x^{2} + 30\,n - 48\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-80\,x^{2} + 64\,x^{3} + 48\,n x^{2}\right)\,P_{n-2}'(x) + \left(-7 + 16\,x + 6\,n\right)\,P_{n-1}(x) + \left(-16\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + 16\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	16														17
2	4	80	144													228
3	28	624	1728	1024												3404
4	280	6688	22896	22528	6400											58792
5	3640	91424	357840	455680	224000	36864										1169448
6	58240	1521024	6548256	9835520	6432000	1880064	200704									26475808
7	1106560	29831296	138106080	233440256	183680000	72769536	14049280	1048576								674031584
8	24344320	673993472	3306815424	6117773312	5499961600	2658926592	705273856	96468992	5308416							19088865984
9	608608000	17239345920	88736326848	176459464704	175735123200	98152943616	32108224512	6096420864	621084672	26214400						595783756736
10	17041024000	492439413760	2639771265024	5571878891520	6023455104000	3760516730880	1433418534912	338448875520	48253501440	3801088000	126877696					20329151306752
11	528271744000	15538278210560	86264863939584	191499952410624	221551351296000	151271120056320	64909899005952	17980795650048	3209255976960	356122624000	22330474496	603979776				753132845368320
12	17961239296000	536753807063040	3072849877840896	7124437414420480	8729620646630400	6419353865379840	3030523775397888	950468157112320	200142481195008	27956346880000	2482869633024	126835752960	2835349504			30112680151951360
13	664565853952000	20147270690068480	118526229743680512	285455561242648576	367523697765452800	287798707943645184	147079195179012096	51000262354010112	12217016849596416	2022742766387200	227174895321088	16512203096064	700331327488	13153337344		1292679650971535360
14	26582634158080000	816523881265971200	4922374625957836800	12261075638994337792	16485045168384665600	13628884816874815488	7450071883742552064	2808424248565039104	746869502151032832	141240658060902400	18877070802157568	1741915724709888	105528873189376	3775007817728	60397977600	59307821408961085440

Technical details

Node id: n-e2824a1907e7ef5d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-4fc9aa67d807dc64

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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;
  1:         1,          16;
  2:         4,          80,         144;
  3:        28,         624,        1728,         1024;
  4:       280,        6688,       22896,        22528,         6400;
  5:      3640,       91424,      357840,       455680,       224000,       36864;
  6:     58240,     1521024,     6548256,      9835520,      6432000,     1880064,      200704;
  7:   1106560,    29831296,   138106080,    233440256,    183680000,    72769536,    14049280,    1048576;
  8:  24344320,   673993472,  3306815424,   6117773312,   5499961600,  2658926592,   705273856,   96468992,   5308416;
  9: 608608000, 17239345920, 88736326848, 176459464704, 175735123200, 98152943616, 32108224512, 6096420864, 621084672, 26214400;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(48*n-80)*x-144*x^2)*P(n-2,x) + ((48*n-80)*x^2+64*x^3)*d/dx P(n-2,x) + (6*n-7+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 16*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 16*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (48*n - 80)*x - 144*x^2)*P[n - 2, x] + ((48*n - 80)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (6*n - 7 + 16*x)*P[n - 1, x] - 16*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (48*n - 80)*x - 144*x^2)*v[i - 2] + ((48*n - 80)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (6*n - 7 + 16*x)*v[i - 1] - 16*x^2*deriv(v[i - 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 (−∞, 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 basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    ↑ Parent-row provers of polytool (above)

References

[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.