A139548 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + x^{2} + 6\,n x - 4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 32\,x + 48\,x^{2} + 32\,x^{3} + 8\,x^{4} + 48\,n x + 128\,n x^{2} + 112\,n x^{3} + 32\,n x^{4} + 32\,n^{2} x^{2} + 64\,n^{2} x^{3} + 32\,n^{2} x^{4}\right)\,P_{n-1}(x)\)

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

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	8	64	72																											144
2	64	1024	3456	4096	1600																									10240
3	512	12288	69120	163840	192000	110592	25088																							573440
4	4096	131072	1032192	3670016	7168000	8257536	5619712	2097152	331776																					28311552
5	32768	1310720	13271040	62914560	172032000	297271296	337182720	251658240	119439360	32768000	3964928																			1291845632
6	262144	12582912	155713536	922746880	3244032000	7474249728	11868831744	13287555072	10510663680	5767168000	2093481984	452984832	44302336																	55834574848
7	2097152	117440512	1717567488	12213813248	52481228800	151145938944	308589625344	460635242496	510117543936	419849830400	254009147392	109924319232	32252100608	5754585088	471859200															2319282339840
8	16777216	1073741824	18119393280	150323855360	763363328000	2638183661568	6583245340672	12283606466560	17489744363520	19193135104000	16256585433088	10552734646272	5160336097280	1841467228160	452984832000	68719476736	4848615424													93458488360960
9	134217728	9663676416	184817811456	1752346656768	10267656192000	41399189766144	122089277227008	273366078455808	475721046687744	652566593536000	710645020360704	615073676525568	421083425538048	225395588726784	92408905728000	28037546508288	5934705278976	782757789696	48452599808											3676766883282944
10	1073741824	85899345920	1836098519040	19585050869760	130056978432000	599302556614656	2039293421813760	5327133836574720	10955999863111680	18034567675904000	24004009576628224	25969777453301760	22858814529208320	16314347374510080	9364102447104000	4261707069259776	1503458670673920	396597280112640	73647951708160	8589934592000	473520144384									141863388262170624
11	8589934592	755914244096	17858474016768	211655988346880	1570884288512000	8143464151646208	31405118695931904	93757555523715072	222491074143191040	427280526475264000	672112268145590272	872584522430939136	938735316666155008	837469831891517440	618030761508864000	375030222094860288	185226108227026944	73291177364815872	22683569126113280	5291399708672000	875065226821632	91465623535616	4544075399168							5386305154335113216
12	68719476736	6597069766656	170699180212224	2225411534618624	18255328994918400	105150695010729984	453218967847174144	1522181489679138816	4093835764234715136	8985099071022694400	16307954813906190336	24700854173429661696	31404963321195003904	33620607069390372864	30324709364701593600	23001853621818097664	14606401677331267584	7706043791500640256	3339021375363874816	1168341055674777600	322024003470360576	67318698922213376	10033318481362944	949978046398464	42949672960000					201761263306198220800
13	549755813888	57174604644352	1608035755622400	22869841857740800	205471235440640000	1301865747751895040	6201943770540277760	23144279960033689600	69568450895491891200	171773952828375040000	353339020967967457280	611640198580163051520	897284666319857254400	1120686902313012428800	1194609762851880960000	1087360353031400980480	843925430245806571520	556547607163935129600	310051984855216947200	144651749750210560000	55817493934862499840	17502861719775477760	4347771341923942400	823314306878668800	111669149696000000	9662508184895488	400771988324352			7470931349852368404480
14	4398046511104	492581209243648	14962154230775808	230528005926027264	2251250057871360000	15560640400006840320	81189082087072727040	333277631424485130240	1107236818462986731520	3037686744754421760000	6983641826190415626240	13599999709605978439680	22611573591260402810880	32275782786614757949440	39697801350154813440000	42156124455986622627840	38667128803989682913280	30599999346613451489280	20835493382270578851840	12150746979017687040000	6028289344965149982720	2520412087647668797440	876510702531866787840	248970246400109445120	56281251446784000000	9739808250374651904	1211934492692840448	96545917011755008	3698757115838464	273897256006439422394368

Technical details

Node id: n-eac81999dec1bbfd · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-561a7d876fc231a6

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:         8,         64,           72;
  2:        64,       1024,         3456,          4096,           1600;
  3:       512,      12288,        69120,        163840,         192000,         110592,           25088;
  4:      4096,     131072,      1032192,       3670016,        7168000,        8257536,         5619712,         2097152,          331776;
  5:     32768,    1310720,     13271040,      62914560,      172032000,      297271296,       337182720,       251658240,       119439360,        32768000,         3964928;
  6:    262144,   12582912,    155713536,     922746880,     3244032000,     7474249728,     11868831744,     13287555072,     10510663680,      5767168000,      2093481984,       452984832,        44302336;
  7:   2097152,  117440512,   1717567488,   12213813248,    52481228800,   151145938944,    308589625344,    460635242496,    510117543936,    419849830400,    254009147392,    109924319232,     32252100608,      5754585088,      471859200;
  8:  16777216, 1073741824,  18119393280,  150323855360,   763363328000,  2638183661568,   6583245340672,  12283606466560,  17489744363520,  19193135104000,  16256585433088,  10552734646272,   5160336097280,   1841467228160,   452984832000,    68719476736,    4848615424;
  9: 134217728, 9663676416, 184817811456, 1752346656768, 10267656192000, 41399189766144, 122089277227008, 273366078455808, 475721046687744, 652566593536000, 710645020360704, 615073676525568, 421083425538048, 225395588726784, 92408905728000, 28037546508288, 5934705278976, 782757789696, 48452599808;

%F Formula

The row polynomials satisfy (1+(6*n-4)*x+(4*n^2-4*n+1)*x^2)*P(n,x) = (8+(48*n+32)*x+(32*n^2+128*n+48)*x^2+(64*n^2+112*n+32)*x^3+(32*n^2+32*n+8)*x^4)*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[((8 + (48*n + 32)*x + (32*n^2 + 128*n + 48)*x^2 + (64*n^2 + 112*n + 32)*x^3 + (32*n^2 + 32*n + 8)*x^4)*P[n - 1, x])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*x^2)]];
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] = ((8 + (48*n + 32)*x + (32*n^2 + 128*n + 48)*x^2 + (64*n^2 + 112*n + 32)*x^3 + (32*n^2 + 32*n + 8)*x^4)*v[i - 1])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*x^2)); 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 T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from basis_map_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    
    ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

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.