A132166 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in [−37, −1)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(50 + 2\,x + 6\,n\right)\,P_{n-1}(x) + \left(37 + 38\,x + x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 379 + 44\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	379	44	1															424
1	5713	1242	66	1														7022
2	80032	28102	2589	88	1													110812
3	1080325	555077	77815	4420	110	1												1717748
4	14354557	10012512	1975443	165500	6735	132	1											26514880
5	189853438	169546039	44572143	5125694	301805	9534	154	1										409408808
6	2511177679	2744639366	922030813	138677168	11024650	497378	12817	176	1									6328060048
7	33258000301	43005575310	17852826033	3388726806	347541537	20925387	762867	16584	198	1								97874375024
8	440953665088	657853033564	328323375558	76477957680	9790928625	752828832	36315237	1108920	20835	220	1							1514189234560
9	5849978617585	9882747323963	5796735000301	1619545476081	252345070860	24011602581	1467805899	58915788	1546185	25570	242	1						23426891385056
10	77627638378537	146408073169536	99046961528682	32562476702356	6050329417170	695646469926	52266061617	2642496480	90682884	2085310	30789	264	1					362446127023552
11	1030136054227366	2145112406999149	1647905410498362	627214731528412	136642518874975	18626632998525	1680961219575	103886020140	4468834845	133806625	2736943	36492	286	1				5607427207781696
12	13669717178313811	31148217173679824	26822762913288457	11655826071746332	2934803395607489	466969199969020	49708622470479	3671882712336	192217028757	7185819424	190711367	3511732	42679	308	1			86751876034902016
13	181387989640645081	448926746298176514	428692811237232972	210146991472854659	60400836445193580	11071271259514977	1369754476360543	118830438058518	7404848668890	335629288915	11084666439	264055722	4420325	49350	330	1		1342122983099186816
14	2406840013228174816	6429496727889581794	6746975685423548283	3692385230829737480	1198444668452935760	250210729441532622	35536888395075151	3569532131781352	260268365453481	13994667929110	558641847445	16513963536	356732558	5473370	56505	352	1	20763734314343823616

Technical details

Node id: n-6cf51bee0786f309 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-67df8a2168740ae3

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:           379,            44,             1;
  1:          5713,          1242,            66,             1;
  2:         80032,         28102,          2589,            88,            1;
  3:       1080325,        555077,         77815,          4420,          110,           1;
  4:      14354557,      10012512,       1975443,        165500,         6735,         132,          1;
  5:     189853438,     169546039,      44572143,       5125694,       301805,        9534,        154,        1;
  6:    2511177679,    2744639366,     922030813,     138677168,     11024650,      497378,      12817,      176,       1;
  7:   33258000301,   43005575310,   17852826033,    3388726806,    347541537,    20925387,     762867,    16584,     198,     1;
  8:  440953665088,  657853033564,  328323375558,   76477957680,   9790928625,   752828832,   36315237,  1108920,   20835,   220,   1;
  9: 5849978617585, 9882747323963, 5796735000301, 1619545476081, 252345070860, 24011602581, 1467805899, 58915788, 1546185, 25570, 242, 1;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = (6*n+50+2*x)*P(n-1,x) + (37+38*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 379 + 44*x + x^2.

%t Mathematica

P[0, x_] := 379 + 44*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 50 + 2*x)*P[n - 1, x] + (37 + 38*x + 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] = 379 + 44*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 50 + 2*x)*v[i - 1] + (37 + 38*x + 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 [-37, -1)
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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

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