A272867 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 18\,x - 36\,x^{2} - 12\,x^{3} + 4\,x^{4} + 54\,n x + 48\,n x^{2} - 12\,n x^{3} - 16\,n x^{4} + 36\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n}(x) = \left(54 + 324\,x + 756\,x^{2} + 864\,x^{3} + 504\,x^{4} + 144\,x^{5} + 16\,x^{6} + 324\,n x + 1368\,n x^{2} + 2160\,n x^{3} + 1584\,n x^{4} + 528\,n x^{5} + 64\,n x^{6} + 216\,n^{2} x^{2} + 720\,n^{2} x^{3} + 816\,n^{2} x^{4} + 384\,n^{2} x^{5} + 64\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

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

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	6	48	36																											90
2	36	576	1728	1536	400																									4276
3	216	5184	27216	55296	50400	20736	3136																							162184
4	1296	41472	311040	995328	1641600	1492992	752640	196608	20736																					5453712
5	7776	311040	3032640	13271040	31968000	46282752	41771520	23592960	8087040	1536000	123904																			169984672
6	46656	2239488	26873856	149299200	474336000	949542912	1257209856	1125384192	683043840	276480000	71368704	10616832	692224																	5027133760
7	279936	15676416	223388928	1504935936	5943974400	15237476352	26771254272	33209450496	29503014912	18811699200	8524099584	2675441664	552394752	67436544	3686400															143044209792
8	1679616	107495424	1773674496	14046068736	66363494400	207681159168	456107065344	727210524672	858404708352	757510963200	500580089856	246140633088	88615747584	22658678784	3892838400	402653184	18939904													3951516414208
9	10077696	725594112	13604889600	123834728448	681252249600	2519262756864	6649057787904	12988887072768	19210829709312	21807238348800	19131772796928	12988887072768	6794804330496	2709330591744	807410073600	173946175488	25568870400	2293235712	94633984											106628810996224
10	60466176	4837294080	101583175680	1044855521280	6560580096000	28002127970304	86383322726400	200061883514880	356498899107840	496628858880000	546222374387712	476763704524800	330580597800960	181537635041280	78352220160000	26222385954816	6658133852160	1238347284480	158985093120	12582912000	462422016									2823034438185984
11	362797056	31926140928	742282776576	8513637580800	60083223552000	290783291572224	1027102974492672	2753423908798464	5748620935495680	9511625097216000	12618576229957632	13521761668694016	11749528280236032	8285682306908160	4731375255552000	2175544816828416	797736483422208	229754699513856	50778703134720	8304721920000	946115444736	66991423488	2218786816							73570986132244480
12	2176782336	208971104256	5328763158528	67428009639936	528377632358400	2859560590639104	11383130748616704	34686602777198592	83100807630618624	159443404613222400	248170038146629632	316123313426399232	331425816196939776	286852636257288192	205099205079859200	120940775054770176	58567007237308416	23124401851465728	7362499728900096	1859593332326400	363674569015296	53057207402496	5431590125568	347892350976	10485760000					1892022659969880064
13	13060694016	1358312177664	37693162930176	521591876222976	4493749454438400	26894581117747200	119147421443162112	406867421077438464	1100675673677758464	2404574771045990400	4300691437854130176	6359248154910523392	7827095520943276032	8054737668786880512	6947126202020659200	5024591134744117248	3043525743593127936	1538927853469433856	645989197903036416	223246870275686400	62760781665533952	14108393786572800	2473450291593216	325627240513536	30261903360000	1769258090496	48922361856			48108094391797456896
14	78364164096	8776786378752	263303591362560	3955405061357568	37161157327257600	243713804165185536	1189393682638307328	4499908343699079168	13571184929222688768	33276676284717465600	67295451167651266560	113423978491778433024	160548988045520535552	191881938148251402240	194323635371822284800	167080871312763125760	122021453110204956672	75615985661185622016	39659068048602562560	17528454997711257600	6486699579034042368	1991728978829770752	501020097352040448	101432885419966464	16110783037440000	1932029834821632	164379135836160	8839042695168	225754218496	1211301187918505525248

Technical details

Node id: n-a9ac7753e25a7e6b · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-9c6b6869d5c10ef7

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:        6,        48,          36;
  2:       36,       576,        1728,         1536,          400;
  3:      216,      5184,       27216,        55296,        50400,         20736,          3136;
  4:     1296,     41472,      311040,       995328,      1641600,       1492992,        752640,         196608,          20736;
  5:     7776,    311040,     3032640,     13271040,     31968000,      46282752,      41771520,       23592960,        8087040,        1536000,         123904;
  6:    46656,   2239488,    26873856,    149299200,    474336000,     949542912,    1257209856,     1125384192,      683043840,      276480000,       71368704,       10616832,        692224;
  7:   279936,  15676416,   223388928,   1504935936,   5943974400,   15237476352,   26771254272,    33209450496,    29503014912,    18811699200,     8524099584,     2675441664,     552394752,      67436544,      3686400;
  8:  1679616, 107495424,  1773674496,  14046068736,  66363494400,  207681159168,  456107065344,   727210524672,   858404708352,   757510963200,   500580089856,   246140633088,   88615747584,   22658678784,   3892838400,    402653184,    18939904;
  9: 10077696, 725594112, 13604889600, 123834728448, 681252249600, 2519262756864, 6649057787904, 12988887072768, 19210829709312, 21807238348800, 19131772796928, 12988887072768, 6794804330496, 2709330591744, 807410073600, 173946175488, 25568870400, 2293235712, 94633984;

%F Formula

The row polynomials satisfy (9+(54*n-18)*x+(36*n^2+48*n-36)*x^2+(48*n^2-12*n-12)*x^3+(16*n^2-16*n+4)*x^4)*P(n,x) = (54+(324*n+324)*x+(216*n^2+1368*n+756)*x^2+(720*n^2+2160*n+864)*x^3+(816*n^2+1584*n+504)*x^4+(384*n^2+528*n+144)*x^5+(64*n^2+64*n+16)*x^6)*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[Together[((54 + (324*n + 324)*x + (216*n^2 + 1368*n + 756)*x^2 + (720*n^2 + 2160*n + 864)*x^3 + (816*n^2 + 1584*n + 504)*x^4 + (384*n^2 + 528*n + 144)*x^5 + (64*n^2 + 64*n + 16)*x^6)*P[n - 1, x])/(9 + (54*n - 18)*x + (36*n^2 + 48*n - 36)*x^2 + (48*n^2 - 12*n - 12)*x^3 + (16*n^2 - 16*n + 4)*x^4)]];
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] = ((54 + (324*n + 324)*x + (216*n^2 + 1368*n + 756)*x^2 + (720*n^2 + 2160*n + 864)*x^3 + (816*n^2 + 1584*n + 504)*x^4 + (384*n^2 + 528*n + 144)*x^5 + (64*n^2 + 64*n + 16)*x^6)*v[i - 1])/(9 + (54*n - 18)*x + (36*n^2 + 48*n - 36)*x^2 + (48*n^2 - 12*n - 12)*x^3 + (16*n^2 - 16*n + 4)*x^4)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013609 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)

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.