A225472 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−2, −1)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(5 + 3\,x\right)\,P_{n-1}(x) + \left(6 + 9\,x + 3\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	3														8
3	43	57	18													118
4	557	1143	756	162												2618
5	9643	26745	27162	11988	1944											77482
6	208685	729303	1000980	674730	223560	29160										2866418
7	5419243	22848057	39543498	35965620	18137520	4811400	524880									127250218
8	164184557	810652503	1693969956	1942091802	1319567760	531528480	117573120	11022480								6590590658
9	5684837803	32169328185	78782876442	109062293508	93353234184	50602572720	16967270880	3218564160	264539520							390105517402
10	221440158125	1412819625303	3968143410420	6439516900650	6654307305240	4541248900440	2047094985600	587872948320	97615082880	7142567040						25977201884018
11	9584118542443	68061516153657	215655520403178	401481206840340	486338269335840	400572474082200	227211748187760	87649879161600	22011010761600	3249868003200	214277011200					1922029888483018
12	456289689634157	3569479826616663	12595107789321156	26460416766579642	36774817747319520	35503320678906960	24297092601344640	11788167011488560	3973937800694400	886642559510400	117852356160000	7071141369600				156430195968945698
13	23698327407870763	202482880113389625	787391859912520122	1842684932570107428	2890392149206264824	3201483633803547840	2567691250182178560	1502583315842790720	636787917142755840	190618722106243200	38261945950905600	4624526455718400	254561089305600			13888956021783598522
14	1333388917719691565	12354557622761308503	52493238844930234260	135421856162406474570	236662806322404582120	295866948934829792520	272228988434609606400	186663001753987799040	95388080193956309760	35882200621045658880	9658665274600742400	1761923366204601600	195248355497395200	9927882482918400		1335920832687437115218
15	80794290325166308843	806882839610980204857	3719065250093279781258	10485845898539089265460	20204245801212359337360	28143533125223504188200	29226961472136390602640	22988520725707874937600	13762433050566808608000	6240867335122791571200	2110326161887944000000	516033865742915520000	86266680188238950400	8825887527314457600	416971064282572800	138381019354948940306218

Technical details

Node id: n-6c1b3ca24c1db31c · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-bba9e798c16ddeb5

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:            5,             3;
   3:           43,            57,            18;
   4:          557,          1143,           756,           162;
   5:         9643,         26745,         27162,         11988,          1944;
   6:       208685,        729303,       1000980,        674730,        223560,         29160;
   7:      5419243,      22848057,      39543498,      35965620,      18137520,       4811400,        524880;
   8:    164184557,     810652503,    1693969956,    1942091802,    1319567760,     531528480,     117573120,     11022480;
   9:   5684837803,   32169328185,   78782876442,  109062293508,   93353234184,   50602572720,   16967270880,   3218564160,   264539520;
  10: 221440158125, 1412819625303, 3968143410420, 6439516900650, 6654307305240, 4541248900440, 2047094985600, 587872948320, 97615082880, 7142567040;

%F Formula

The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) + (6+9*x+3*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 3*x)*P[n - 1, x] + (6 + 9*x + 3*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (5 + 3*x)*v[i - 1] + (6 + 9*x + 3*x^2)*deriv(v[i - 1], x)); 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 (-2, -1)
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
  - 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