A253283 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(126 - 253\,n + 167\,n^{2} - 44\,n^{3} + 4\,n^{4}\right)\,P_{n}(x) = \left(-300 + 455\,n - 239\,n^{2} + 52\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(1104 + 2160\,x - 1684\,n - 3324\,n x + 896\,n^{2} + 1784\,n^{2} x - 200\,n^{3} - 400\,n^{3} x + 16\,n^{4} + 32\,n^{4} x\right)\,P_{n-3}(x) + \left(-1458 - 3888\,x - 3780\,x^{2} + 2268\,n + 6048\,n x + 5952\,n x^{2} - 1242\,n^{2} - 3312\,n^{2} x - 3296\,n^{2} x^{2} + 288\,n^{3} + 768\,n^{3} x + 768\,n^{3} x^{2} - 24\,n^{4} - 64\,n^{4} x - 64\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(792 + 1512\,x - 1292\,n - 2532\,n x + 752\,n^{2} + 1496\,n^{2} x - 184\,n^{3} - 368\,n^{3} x + 16\,n^{4} + 32\,n^{4} x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 2\,x\)

\(P_{3}(x) = 4\,x + 9\,x^{2}\)

\(P_{4}(x) = 6\,x + 36\,x^{2} + 40\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	0	2							2
3	0	4	9						13
4	0	6	36	40					82
5	0	8	90	240	175				513
6	0	10	180	840	1400	756			3186
7	0	12	315	2240	6300	7560	3234		19661
8	0	14	504	5040	21000	41580	38808	13728	120674

All 15 rows

Technical details

Node id: n-bd89bcbd060a3e48 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c9871221535ca904

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 0,  2;
   3: 0,  4,    9;
   4: 0,  6,   36,    40;
   5: 0,  8,   90,   240,    175;
   6: 0, 10,  180,   840,   1400,    756;
   7: 0, 12,  315,  2240,   6300,   7560,    3234;
   8: 0, 14,  504,  5040,  21000,  41580,   38808,   13728;
   9: 0, 16,  756, 10080,  57750, 166320,  252252,  192192,  57915;
  10: 0, 18, 1080, 18480, 138600, 540540, 1177176, 1441440, 926640, 243100;

%F Formula

The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+896*n^2-1684*n+1104+(32*n^4-400*n^3+1784*n^2-3324*n+2160)*x)*P(n-3,x) - (24*n^4-288*n^3+1242*n^2-2268*n+1458+(64*n^4-768*n^3+3312*n^2-6048*n+3888)*x+(64*n^4-768*n^3+3296*n^2-5952*n+3780)*x^2)*P(n-2,x) + (16*n^4-184*n^3+752*n^2-1292*n+792+(32*n^4-368*n^3+1496*n^2-2532*n+1512)*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2*x, P(3,x) = 4*x + 9*x^2, P(4,x) = 6*x + 36*x^2 + 40*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2*x;
P[3, x_] := 4*x + 9*x^2;
P[4, x_] := 6*x + 36*x^2 + 40*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*P[n - 3, x] - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (64*n^4 - 768*n^3 + 3312*n^2 - 6048*n + 3888)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*P[n - 2, x] + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*P[n - 1, x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; v[3] = 4*x + 9*x^2; v[4] = 6*x + 36*x^2 + 40*x^3; for(n = 5, N, my(i = n); v[i] = (-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*v[i - 4] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*v[i - 3] - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (64*n^4 - 768*n^3 + 3312*n^2 - 6048*n + 3888)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*v[i - 2] + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*v[i - 1])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/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

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.