A272867 → x ↦ x+1 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(13 + 16\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	13	16	4																											33
2	169	416	360	128	16																									1089
3	2197	8112	12012	9088	3696	768	64																							35937
4	28561	140608	294736	342784	241504	105472	27904	4096	256																					1185921
5	371293	2284880	6195540	9734400	9803040	6606336	3016320	921600	180480	20480	1024																			39135393
6	4826809	35644128	118585272	234815360	307972080	281668608	184125696	86667264	29157120	6840320	1062912	98304	4096																	1291467969
7	62748517	540602608	2131221820	5092540544	8235023888	9528506624	8132220096	5199360000	2502221568	902107136	239891456	45645824	5877760	458752	16384															42618442977
8	815730721	8031810176	36606519456	102464986624	197060846528	276001130496	291115062784	235821228032	148247520768	72560377856	27561189376	8040087552	1766309888	282591232	31064064	2097152	65536													1406408618241
9	10604499373	117465223824	607656638628	1949876378112	4347656868672	7150848187392	8988757290240	8827521490944	6864817669632	4258530156544	2112251590656	835741679616	261848186880	64094994432	11990532096	1654652928	158662656	9437184	262144											46411484401953
10	137858491849	1696719899680	9821397880840	35540760028800	90148187897040	170323041847296	248658043246080	287181288775680	266438002721280	200508060712960	123055023861760	61694787911680	25224899665920	8365772636160	2228789575680	469738979328	76499189760	9279897600	789053440	41943040	1048576									1531578985264449
11	1792160394037	24263094565424	155377124813196	625959126066560	1779864194645680	3798733590482688	6318315983343936	8393177613410304	9053226828771840	8018337987911680	5873596292495360	3572944867491840	1807260397690880	759132590571520	263726225326080	75230330486784	17425471438848	3223581622272	464732815360	50289704960	3840933888	184549376	4194304							50542106513726817
12	23298085122481	344094795655104	2415280777194480	10720555014138112	33775089046711584	80365200294872064	150037302009776896	225399299069767680	277256054521974528	282662733556842496	240863066924113920	172499175908966400	104155888219832320	53076669510451200	22803603969146880	8234144263831552	2485121317797888	621635426058240	127320718311424	20983817699328	2713498681344	265012903936	18371051520	805306368	16777216					1667889514952984961
13	302875106592253	4846001705476048	36997359174499828	179388086801527552	620266160942238304	1628031248637274624	3371428487031899008	5652248521242898432	7810866701941059328	9012309604869615616	8762847825010859008	7226949291829755904	5077465629097738240	3046487618789048320	1562297116645457920	684208216977965056	255267301229264896	80779778678849536	21541821426171904	4796462827307008	880299439292416	130796417449984	15333024858112	1364457422848	86587211776	3489660928	67108864			55040353993448503713
14	3937376385699289	67844023876664672	559713196982483544	2943386882033759616	11081658917771538096	31806217154566493184	72358134953380034304	133934211563217162240	205462957413247742208	264742886179333545984	289357442210819254272	270205144413439033344	216689633147590127616	149751586277140463616	89363526933406679040	46077411162197065728	20514994854209716224	7871246810405142528	2593589342318886912	730142274289139712	174354583652401152	34970995764756480	5813269759524864	786252014026752	84289051951104	6888590671872	403055837184	15032385536	268435456	1816331681783800622529

Technical details

Node id: n-43ea2eaea8001298 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-befc9532e898cdfb

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:          13,           16,            4;
  2:         169,          416,          360,           128,            16;
  3:        2197,         8112,        12012,          9088,          3696,           768,            64;
  4:       28561,       140608,       294736,        342784,        241504,        105472,         27904,          4096,           256;
  5:      371293,      2284880,      6195540,       9734400,       9803040,       6606336,       3016320,        921600,        180480,         20480,          1024;
  6:     4826809,     35644128,    118585272,     234815360,     307972080,     281668608,     184125696,      86667264,      29157120,       6840320,       1062912,        98304,         4096;
  7:    62748517,    540602608,   2131221820,    5092540544,    8235023888,    9528506624,    8132220096,    5199360000,    2502221568,     902107136,     239891456,     45645824,      5877760,      458752,       16384;
  8:   815730721,   8031810176,  36606519456,  102464986624,  197060846528,  276001130496,  291115062784,  235821228032,  148247520768,   72560377856,   27561189376,   8040087552,   1766309888,   282591232,    31064064,    2097152,     65536;
  9: 10604499373, 117465223824, 607656638628, 1949876378112, 4347656868672, 7150848187392, 8988757290240, 8827521490944, 6864817669632, 4258530156544, 2112251590656, 835741679616, 261848186880, 64094994432, 11990532096, 1654652928, 158662656, 9437184, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (13+16*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(13 + 16*x + 4*x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = (13 + 16*x + 4*x^2)*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.