A123160 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{3}(x) = 5 + 10\,x + 3\,x^{2}\)

\(P_{4}(x) = 10 + 45\,x + 48\,x^{2} + 10\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	5	10	3													18
4	10	45	48	10												113
5	17	136	300	220	35											708
6	26	325	1200	1750	980	126										4407
7	37	666	3675	8680	9450	4284	462									27254
8	50	1225	9408	32340	55860	48510	18480	1716								167589
9	65	2080	21168	99120	244860	332640	240240	78936	6435							1025544
10	82	3321	43200	263340	873180	1675674	1873872	1158300	334620	24310						6249899
11	101	5050	81675	627000	2672670	6810804	10660650	10124400	5469750	1409980	92378					37954458
12	122	7381	145200	1368510	7267260	23585562	48624576	64178400	52947180	25403950	5912192	352716				229793049
13	145	10440	245388	2782780	17972955	72072000	187867680	324509328	369973890	269743760	116396280	24690120	1352078			1387616844
14	170	14365	397488	5335330	41121080	199098900	637116480	1383452928	2053562940	2059567510	1345023680	527310420	102757928	5200300		8359959519
15	197	19306	621075	9733360	88148060	506017512	1943817876	5154692400	9573594030	12447935500	11139770642	6587324016	2366136500	426424600	20058300	50264293374

Technical details

Node id: n-cc7078141af78a67 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-7537db165a517602

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:  2,    1;
   3:  5,   10,     3;
   4: 10,   45,    48,     10;
   5: 17,  136,   300,    220,     35;
   6: 26,  325,  1200,   1750,    980,     126;
   7: 37,  666,  3675,   8680,   9450,    4284,     462;
   8: 50, 1225,  9408,  32340,  55860,   48510,   18480,    1716;
   9: 65, 2080, 21168,  99120, 244860,  332640,  240240,   78936,   6435;
  10: 82, 3321, 43200, 263340, 873180, 1675674, 1873872, 1158300, 334620, 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+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) = 1, P(2,x) = 2 + x, P(3,x) = 5 + 10*x + 3*x^2, P(4,x) = 10 + 45*x + 48*x^2 + 10*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[3, x_] := 5 + 10*x + 3*x^2;
P[4, x_] := 10 + 45*x + 48*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 + 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] = 1; v[2] = 2 + x; v[3] = 5 + 10*x + 3*x^2; v[4] = 10 + 45*x + 48*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 + 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

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.