A123160 → D

nonnegative coefficients

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

Recurrence

\(\left(126 - 253\,n + 167\,n^{2} - 44\,n^{3} + 4\,n^{4}\right)\,P_{n}(x) = \left(-300 + 455\,n - 239\,n^{2} + 52\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(1056 + 2160\,x - 1640\,n - 3324\,n x + 888\,n^{2} + 1784\,n^{2} x - 200\,n^{3} - 400\,n^{3} x + 16\,n^{4} + 32\,n^{4} x\right)\,P_{n-3}(x) + \left(-1350 - 3672\,x - 3780\,x^{2} + 2172\,n + 5856\,n x + 5952\,n x^{2} - 1226\,n^{2} - 3280\,n^{2} x - 3296\,n^{2} x^{2} + 288\,n^{3} + 768\,n^{3} x + 768\,n^{3} x^{2} - 24\,n^{4} - 64\,n^{4} x - 64\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(720 + 1512\,x - 1240\,n - 2532\,n x + 744\,n^{2} + 1496\,n^{2} x - 184\,n^{3} - 368\,n^{3} x + 16\,n^{4} + 32\,n^{4} x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1\)

\(P_{3}(x) = 4 + 6\,x\)

\(P_{4}(x) = 9 + 36\,x + 30\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	1							1
3	4	6						10
4	9	36	30					75
5	16	120	240	140				516
6	25	300	1050	1400	630			3405
7	36	630	3360	7560	7560	2772		21918
8	49	1176	8820	29400	48510	38808	12012	138775

All 15 rows

Technical details

Node id: n-4cab0236cf9d2d4f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-2c1fea4a558d4e6c

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:  0;
   2:  1;
   3:  4,    6;
   4:  9,   36,    30;
   5: 16,  120,   240,    140;
   6: 25,  300,  1050,   1400,    630;
   7: 36,  630,  3360,   7560,   7560,    2772;
   8: 49, 1176,  8820,  29400,  48510,   38808,   12012;
   9: 64, 2016, 20160,  92400, 221760,  288288,  192192,  51480;
  10: 81, 3240, 41580, 249480, 810810, 1513512, 1621620, 926640, 218790;

%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+888*n^2-1640*n+1056+(32*n^4-400*n^3+1784*n^2-3324*n+2160)*x)*P(n-3,x) - (24*n^4-288*n^3+1226*n^2-2172*n+1350+(64*n^4-768*n^3+3280*n^2-5856*n+3672)*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+744*n^2-1240*n+720+(32*n^4-368*n^3+1496*n^2-2532*n+1512)*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 4 + 6*x, P(4,x) = 9 + 36*x + 30*x^2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 4 + 6*x;
P[4, x_] := 9 + 36*x + 30*x^2;
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 + 888*n^2 - 1640*n + 1056 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*P[n - 3, x] - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (64*n^4 - 768*n^3 + 3280*n^2 - 5856*n + 3672)*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 + 744*n^2 - 1240*n + 720 + (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] = 0; v[2] = 1; v[3] = 4 + 6*x; v[4] = 9 + 36*x + 30*x^2; 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 + 888*n^2 - 1640*n + 1056 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*v[i - 3] - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (64*n^4 - 768*n^3 + 3280*n^2 - 5856*n + 3672)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*v[i - 2] + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.?
  - 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

Related ?

Same fingerprint ?

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.