A132166 → reverse

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(3 + 84\,x + n + 42\,n x\right)\,P_{n-1}(x) + \left(-x - 36\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 42\,x + 336\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	42	336															379
1	1	63	1113	4536														5713
2	1	84	2331	23184	54432													80032
3	1	105	3990	65205	412272	598752												1080325
4	1	126	6090	139860	1518048	6531840	6158592											14354557
5	1	147	8631	256410	4010769	30912840	94618368	60046272										189853438
6	1	168	11613	424116	8723925	99302112	567362016	1274921856	560431872									2511177679
7	1	189	15036	652239	16685487	253790901	2193126624	9573741216	16176101760	5043886848								33258000301
8	1	210	18900	950040	29117907	557085438	6534131760	44153890560	150628802688	195030291456	44019376128							440953665088
9	1	231	23205	1326780	47438118	1096591041	16358573673	152347155480	822902685696	2232875211840	2250184914432	374164697088						5849978617585
10	1	252	27951	1791720	73257534	1988496216	36138885615	432719396208	3269692046880	14359883150592	31436007144192	24982689005568	3108445175808					77627638378537
11	1	273	33138	2354121	108382050	3381856758	72635764284	1068204859029	10488403911888	65384283132000	236688434857728	423003242559744	268115731513344	25311625003008				1030136054227366
12	1	294	38766	3023244	154812042	5462679852	135567962388	2371076212392	28822377334464	235921791490560	1229659528075776	3710465074520064	5468064651067392	2791778491072512	202493000024064			13669717178313811
13	1	315	44835	3808350	214742367	8458008174	238367847330	4841551396260	70427864184993	719294099301288	4973126585285376	21910024872066240	55632567877622784	68190758524035072	28292068847112192	1594632375189504		181387989640645081
14	1	336	51345	4718700	290562363	12640003992	399022726014	9241838373240	156900139064361	1929332508320448	16772131332579168	99004449637731456	372078217886964480	801445842630918144	823281820320264192	279779813592072192	12381851383824384	2406840013228174816

Technical details

Node id: n-ab60eb3242173f47 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-f34dcee26c4e5272

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1,  42,   336;
  1: 1,  63,  1113,    4536;
  2: 1,  84,  2331,   23184,    54432;
  3: 1, 105,  3990,   65205,   412272,     598752;
  4: 1, 126,  6090,  139860,  1518048,    6531840,     6158592;
  5: 1, 147,  8631,  256410,  4010769,   30912840,    94618368,     60046272;
  6: 1, 168, 11613,  424116,  8723925,   99302112,   567362016,   1274921856,    560431872;
  7: 1, 189, 15036,  652239, 16685487,  253790901,  2193126624,   9573741216,  16176101760,    5043886848;
  8: 1, 210, 18900,  950040, 29117907,  557085438,  6534131760,  44153890560, 150628802688,  195030291456,   44019376128;
  9: 1, 231, 23205, 1326780, 47438118, 1096591041, 16358573673, 152347155480, 822902685696, 2232875211840, 2250184914432, 374164697088;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = (n+3+(42*n+84)*x)*P(n-1,x) - (x+36*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 42*x + 336*x^2.

%t Mathematica

P[0, x_] := 1 + 42*x + 336*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 3 + (42*n + 84)*x)*P[n - 1, x] - (x + 36*x^2)*D[P[n - 1, x], x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 42*x + 336*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 3 + (42*n + 84)*x)*v[i - 1] - (x + 36*x^2)*deriv(v[i - 1], x))/(n + 3)); 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from A convolution triangle of numbers obtained from A036224.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

Related ?

Same fingerprint ?

A convolution triangle of numbers obtained from A036224.

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741