basis-map coefficients of A297191, shifted → basis A008288 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 17\,x - 35\,x^{2} - 11\,x^{3} + 3\,x^{4} - 8\,n + 88\,n x + 120\,n x^{2} - 24\,n x^{3} - 16\,n x^{4} + 4\,n^{2} - 120\,n^{2} x - 32\,n^{2} x^{2} + 152\,n^{2} x^{3} + 28\,n^{2} x^{4} + 48\,n^{3} x - 112\,n^{3} x^{2} - 176\,n^{3} x^{3} - 16\,n^{3} x^{4} + 64\,n^{4} x^{2} + 64\,n^{4} x^{3}\right)\,P_{n}(x) = \left(x^{3} - 5\,x^{4} - 5\,x^{5} + x^{6} + 8\,n x^{3} + 24\,n x^{4} - 8\,n x^{5} + 8\,n x^{6} - 4\,n^{2} x^{2} - 24\,n^{2} x^{3} - 16\,n^{2} x^{4} - 8\,n^{2} x^{5} + 20\,n^{2} x^{6} - 48\,n^{3} x^{3} - 144\,n^{3} x^{4} - 80\,n^{3} x^{5} + 16\,n^{3} x^{6} - 64\,n^{4} x^{4} - 64\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(4 + 47\,x + 253\,x^{2} + 504\,x^{3} + 276\,x^{4} + 49\,x^{5} + 3\,x^{6} - 8\,n - 40\,n x - 40\,n x^{2} + 240\,n x^{3} + 360\,n x^{4} + 120\,n x^{5} + 8\,n x^{6} + 4\,n^{2} - 56\,n^{2} x - 492\,n^{2} x^{2} - 1072\,n^{2} x^{3} - 660\,n^{2} x^{4} - 88\,n^{2} x^{5} - 4\,n^{2} x^{6} + 48\,n^{3} x + 208\,n^{3} x^{2} + 64\,n^{3} x^{3} - 192\,n^{3} x^{4} - 112\,n^{3} x^{5} - 16\,n^{3} x^{6} + 64\,n^{4} x^{2} + 320\,n^{4} x^{3} + 320\,n^{4} x^{4} + 64\,n^{4} x^{5}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 1 + 12\,x + 9\,x^{2}\)

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	12	9													22
2	1	28	117	112	25											283
3	1	44	369	1008	1025	396	49									2892
4	1	60	765	3696	8025	8316	4165	960	81							26069
5	1	76	1305	9200	32225	60588	63161	36800	11745	1900	121					217122
6	1	92	1989	18544	91225	258588	440461	459712	295569	115900	26741	3312	169			1712303
7	1	108	2817	32752	209025	805068	1963969	3112896	3246561	2236300	1011681	294768	52897	5292	225	12974360

All 15 rows

Technical details

Node id: n-2124eaaf0381b900 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-f17030395a2c695a

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,  12,    9;
  2: 1,  28,  117,   112,     25;
  3: 1,  44,  369,  1008,   1025,     396,       49;
  4: 1,  60,  765,  3696,   8025,    8316,     4165,      960,        81;
  5: 1,  76, 1305,  9200,  32225,   60588,    63161,    36800,     11745,      1900,       121;
  6: 1,  92, 1989, 18544,  91225,  258588,   440461,   459712,    295569,    115900,     26741,      3312,      169;
  7: 1, 108, 2817, 32752, 209025,  805068,  1963969,  3112896,   3246561,   2236300,   1011681,    294768,    52897,     5292,     225;
  8: 1, 124, 3789, 52848, 416025, 2041020,  6578005, 14345152,  21524049,  22414300,  16243645,   8164080,  2812329,   647388,   94725,    7936,    289;
  9: 1, 140, 4905, 79856, 749025, 4482540, 18095945, 50909120, 101773665, 146256300, 152032265, 114545520, 62412545, 24404940, 6741225, 1277696, 157505, 11340, 361;

%F Formula

The row polynomials satisfy (4*n^2-8*n+4+(48*n^3-120*n^2+88*n-17)*x+(64*n^4-112*n^3-32*n^2+120*n-35)*x^2+(64*n^4-176*n^3+152*n^2-24*n-11)*x^3-(16*n^3-28*n^2+16*n-3)*x^4)*P(n,x) = -(4*n^2*x^2+(48*n^3+24*n^2-8*n-1)*x^3+(64*n^4+144*n^3+16*n^2-24*n+5)*x^4+(64*n^4+80*n^3+8*n^2+8*n+5)*x^5-(16*n^3+20*n^2+8*n+1)*x^6)*P(n-2,x) + (4*n^2-8*n+4+(48*n^3-56*n^2-40*n+47)*x+(64*n^4+208*n^3-492*n^2-40*n+253)*x^2+(320*n^4+64*n^3-1072*n^2+240*n+504)*x^3+(320*n^4-192*n^3-660*n^2+360*n+276)*x^4+(64*n^4-112*n^3-88*n^2+120*n+49)*x^5-(16*n^3+4*n^2-8*n-3)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2*x^2 + (48*n^3 + 24*n^2 - 8*n - 1)*x^3 + (64*n^4 + 144*n^3 + 16*n^2 - 24*n + 5)*x^4 + (64*n^4 + 80*n^3 + 8*n^2 + 8*n + 5)*x^5 - (16*n^3 + 20*n^2 + 8*n + 1)*x^6)*P[n - 2, x] + (4*n^2 - 8*n + 4 + (48*n^3 - 56*n^2 - 40*n + 47)*x + (64*n^4 + 208*n^3 - 492*n^2 - 40*n + 253)*x^2 + (320*n^4 + 64*n^3 - 1072*n^2 + 240*n + 504)*x^3 + (320*n^4 - 192*n^3 - 660*n^2 + 360*n + 276)*x^4 + (64*n^4 - 112*n^3 - 88*n^2 + 120*n + 49)*x^5 - (16*n^3 + 4*n^2 - 8*n - 3)*x^6)*P[n - 1, x])/(4*n^2 - 8*n + 4 + (48*n^3 - 120*n^2 + 88*n - 17)*x + (64*n^4 - 112*n^3 - 32*n^2 + 120*n - 35)*x^2 + (64*n^4 - 176*n^3 + 152*n^2 - 24*n - 11)*x^3 - (16*n^3 - 28*n^2 + 16*n - 3)*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; v[2] = 1 + 12*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2*x^2 + (48*n^3 + 24*n^2 - 8*n - 1)*x^3 + (64*n^4 + 144*n^3 + 16*n^2 - 24*n + 5)*x^4 + (64*n^4 + 80*n^3 + 8*n^2 + 8*n + 5)*x^5 - (16*n^3 + 20*n^2 + 8*n + 1)*x^6)*v[i - 2] + (4*n^2 - 8*n + 4 + (48*n^3 - 56*n^2 - 40*n + 47)*x + (64*n^4 + 208*n^3 - 492*n^2 - 40*n + 253)*x^2 + (320*n^4 + 64*n^3 - 1072*n^2 + 240*n + 504)*x^3 + (320*n^4 - 192*n^3 - 660*n^2 + 360*n + 276)*x^4 + (64*n^4 - 112*n^3 - 88*n^2 + 120*n + 49)*x^5 - (16*n^3 + 4*n^2 - 8*n - 3)*x^6)*v[i - 1])/(4*n^2 - 8*n + 4 + (48*n^3 - 120*n^2 + 88*n - 17)*x + (64*n^4 - 112*n^3 - 32*n^2 + 120*n - 35)*x^2 + (64*n^4 - 176*n^3 + 152*n^2 - 24*n - 11)*x^3 - (16*n^3 - 28*n^2 + 16*n - 3)*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.?
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.?
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (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.