A253283 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{3}(x) = 2 + 8\,x + 3\,x^{2}\)

\(P_{4}(x) = 3 + 27\,x + 42\,x^{2} + 10\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	2	8	3													13
4	3	27	42	10												82
5	4	64	210	200	35											513
6	5	125	690	1330	910	126										3186
7	6	216	1785	5600	7560	4032	462									19661
8	7	343	3948	18060	38850	40194	17556	1716								120674
9	8	512	7812	48720	150150	243936	204204	75504	6435							737281
10	9	729	14220	115500	478170	1099098	1429428	1003860	321750	24310						4487074
11	10	1000	24255	248160	1321320	4036032	7357350	7962240	4813380	1361360	92378					27217485
12	11	1331	39270	493350	3273270	12738726	30606576	46091760	42664050	22632610	5727436	352716				164621106
13	12	1728	60918	920920	7432425	35747712	108636528	214239168	274362660	221707200	104756652	23984688	1352078			993202689
14	13	2197	91182	1631630	15715700	91279188	340612272	844704432	1409226390	1568116550	1123870748	478635612	100053772	5200300		5979139986
15	14	2744	132405	2766400	31311280	215639424	966089124	2926535040	6110804700	8820251440	8670876214	5581377984	2163324800	416024000	20058300	35925193869

Technical details

Node id: n-1693564fff166958 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b749493f90aeb594

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: 1,   1;
   3: 2,   8,     3;
   4: 3,  27,    42,     10;
   5: 4,  64,   210,    200,     35;
   6: 5, 125,   690,   1330,    910,     126;
   7: 6, 216,  1785,   5600,   7560,    4032,     462;
   8: 7, 343,  3948,  18060,  38850,   40194,   17556,    1716;
   9: 8, 512,  7812,  48720, 150150,  243936,  204204,   75504,   6435;
  10: 9, 729, 14220, 115500, 478170, 1099098, 1429428, 1003860, 321750, 24310;

%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) = 1 + x, P(3,x) = 2 + 8*x + 3*x^2, P(4,x) = 3 + 27*x + 42*x^2 + 10*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 2 + 8*x + 3*x^2;
P[4, x_] := 3 + 27*x + 42*x^2 + 10*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] = 1 + x; v[3] = 2 + 8*x + 3*x^2; v[4] = 3 + 27*x + 42*x^2 + 10*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 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: 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.