basis-map coefficients of A371080, shifted → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 40\,x - 16\,x^{2} + 30\,n - 24\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 8\,x + 6\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 5 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	4														9
2	24	52	16													92
3	184	540	384	64												1172
4	1952	6760	6768	2432	256											18168
5	26496	102376	125200	64320	14080	1024										333496
6	438496	1835424	2571744	1643840	518400	76800	4096									7088800
7	8564384	38148064	59115168	43978816	17454080	3741696	401408	16384								171420000
8	192842688	903346240	1514506432	1262354688	589293824	160219136	24944640	2031616	65536							4649604800
9	4918444480	24029020224	42945741504	39146336256	20661758208	6658089984	1322065920	156598272	10027008	262144						139848344000
10	140150877440	709725690112	1338035210496	1311995247360	763232194560	279979435008	66271285248	10054041600	937820160	48496640	1048576					4620431347200
11	4412841296640	23054539316992	45491225959168	47416963233024	29871931837440	12150126455808	3291341340672	597913485312	71671480320	5406720000	230686720	4194304				166364196006400
12	152149691061760	817043980174848	1677259779408384	1842016641862144	1240761751612416	549399929075712	165804878671872	34618223591424	4986966048768	484759306240	30208425984	1082130432	16777216			6484557908147200
13	5701383526469120	31375979837779456	66692498815627776	76644764220137984	54672990680568832	26004105922848768	8579769923112960	2003498047389696	332874217488384	39002212401152	3138827321344	164395745280	5016387584	67108864		272050175710387200
14	230713604474572800	1297991998307123200	2845882196306753536	3403924454378692608	2552302475745777664	1290833833852186624	459213685528485888	117646517985165312	21932270363148288	2972495076655104	289071856353280	19589811404800	874814373888	23018340352	268435456	12223723091787468800

Technical details

Node id: n-d7007867f0a138dc · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ef507ae4352d333a

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 0)
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;
  1:          5,           4;
  2:         24,          52,          16;
  3:        184,         540,         384,          64;
  4:       1952,        6760,        6768,        2432,         256;
  5:      26496,      102376,      125200,       64320,       14080,       1024;
  6:     438496,     1835424,     2571744,     1643840,      518400,      76800,       4096;
  7:    8564384,    38148064,    59115168,    43978816,    17454080,    3741696,     401408,     16384;
  8:  192842688,   903346240,  1514506432,  1262354688,   589293824,  160219136,   24944640,   2031616,    65536;
  9: 4918444480, 24029020224, 42945741504, 39146336256, 20661758208, 6658089984, 1322065920, 156598272, 10027008, 262144;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(24*n-40)*x+16*x^2)*P(n-2,x) + (6*n-7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (24*n - 40)*x + 16*x^2)*P[n - 2, x] + (6*n - 7 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (24*n - 40)*x + 16*x^2)*v[i - 2] + (6*n - 7 + 8*x)*v[i - 1]); 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    ↑ Parent-row provers of polytool (above)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.