A200536 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 42\,x - 72\,x^{2} - 14\,x^{3} + 4\,x^{4} + 126\,n x + 96\,n x^{2} - 14\,n x^{3} - 16\,n x^{4} + 49\,n^{2} x^{2} + 56\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n}(x) = \left(216 + 756\,x + 1098\,x^{2} + 847\,x^{3} + 366\,x^{4} + 84\,x^{5} + 8\,x^{6} + 756\,n x + 2046\,n x^{2} + 2198\,n x^{3} + 1170\,n x^{4} + 308\,n x^{5} + 32\,n x^{6} + 294\,n^{2} x^{2} + 679\,n^{2} x^{3} + 586\,n^{2} x^{4} + 224\,n^{2} x^{5} + 32\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

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

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	28	18																											52
2	36	336	657	448	100																									1577
3	216	3024	9882	13552	9150	3024	392																							39240
4	1296	24192	110808	228480	258025	171360	67032	14336	1296																					876825
5	7776	181440	1069200	2943360	4662750	4657212	3046330	1308160	356400	56000	3872																			18292500
6	46656	1306368	9412848	32417280	66109500	88188912	80928841	52260096	23799420	7504000	1562352	193536	10816																	363740625
7	279936	9144576	77892192	322092288	803514600	1341394992	1584966642	1367041984	873348966	414010800	144037432	35788032	6019104	614656	28800															6980175000
8	1679616	62705664	616419072	2975035392	8776706400	17597212416	25440090480	27462779904	22591325169	14303531200	6980160880	2606994432	732475744	149976064	21139200	1835008	73984													130320140625
9	10077696	423263232	4716361728	26022850560	88624238400	207440134272	354707561376	459939658752	462959070966	366876048700	230526944802	114984914688	45310338528	13943164032	3282379200	571146240	69249024	5225472	184832											2380412812500
10	60466176	2821754880	35145964800	218215710720	842572368000	2255075956224	4452095884320	6740243435520	8025238971540	7639448586000	5873763470113	3666935321280	1860446345940	764518352640	252386388000	65992163328	13360955520	2020515840	214867200	14336000	451584									42710572265625
11	362797056	18623582208	256406819328	1769240309760	7637423904000	23005555905024	51501321571392	89164385114112	122584952163960	136253869290000	123925350234378	92935071606384	57695273249614	29673064867600	12611620593000	4403179511808	1250011080384	284019208704	50427252480	6741504000	638313984	38162432	1083392							755027578125000
12	2176782336	121899810816	1838292682752	13955270934528	66600091641600	222980149776384	558138455449344	1087653068587008	1695720112753392	2155538049033600	2263863836313288	1982391432997248	1455830707539481	899418335341344	467740462048200	204376940945408	74693356589808	22659438928896	5640603074304	1132856524800	179061974784	21447286784	1829849088	99090432	2560000					13180497978515625
13	13060694016	792348770304	12988860198912	107583355256832	562449062822400	2072008541611008	5734408407856128	12421716885946368	21655521158233248	30986515478318400	36906815815247664	36961593608983680	31344731578487886	22611953342108956	13910383836607050	7301055527700480	3264790257302288	1239460619132736	397177418997216	106496201011200	23598388509696	4245847237632	604654888448	65589608448	5091840000	251969536	5971968			227626975195312500
14	78364164096	5119792054272	90647746818048	813478070845440	4623547659494400	18585662901571584	56352887711527680	134326319668789248	258940431354661056	411883437283718400	548595892354477968	618465853022943744	594813220355367468	490756679861440944	348627802538374425	213662772184572928	113018422539580012	51538821085245312	20206419461276048	6779974276275200	1933863913485504	464491446042624	92726897652480	15107991994368	1957504640000	194016542720	13816147968	629407744	13778944	3894595824462890625

Technical details

Node id: n-e76af746ebfe4ce3 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-b81b61d2759417d6

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,        28,         18;
  2:       36,       336,        657,         448,         100;
  3:      216,      3024,       9882,       13552,        9150,         3024,          392;
  4:     1296,     24192,     110808,      228480,      258025,       171360,        67032,        14336,         1296;
  5:     7776,    181440,    1069200,     2943360,     4662750,      4657212,      3046330,      1308160,       356400,        56000,         3872;
  6:    46656,   1306368,    9412848,    32417280,    66109500,     88188912,     80928841,     52260096,     23799420,      7504000,      1562352,       193536,       10816;
  7:   279936,   9144576,   77892192,   322092288,   803514600,   1341394992,   1584966642,   1367041984,    873348966,    414010800,    144037432,     35788032,     6019104,      614656,      28800;
  8:  1679616,  62705664,  616419072,  2975035392,  8776706400,  17597212416,  25440090480,  27462779904,  22591325169,  14303531200,   6980160880,   2606994432,   732475744,   149976064,   21139200,   1835008,    73984;
  9: 10077696, 423263232, 4716361728, 26022850560, 88624238400, 207440134272, 354707561376, 459939658752, 462959070966, 366876048700, 230526944802, 114984914688, 45310338528, 13943164032, 3282379200, 571146240, 69249024, 5225472, 184832;

%F Formula

The row polynomials satisfy (36+(126*n-42)*x+(49*n^2+96*n-72)*x^2+(56*n^2-14*n-14)*x^3+(16*n^2-16*n+4)*x^4)*P(n,x) = (216+(756*n+756)*x+(294*n^2+2046*n+1098)*x^2+(679*n^2+2198*n+847)*x^3+(586*n^2+1170*n+366)*x^4+(224*n^2+308*n+84)*x^5+(32*n^2+32*n+8)*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[((216 + (756*n + 756)*x + (294*n^2 + 2046*n + 1098)*x^2 + (679*n^2 + 2198*n + 847)*x^3 + (586*n^2 + 1170*n + 366)*x^4 + (224*n^2 + 308*n + 84)*x^5 + (32*n^2 + 32*n + 8)*x^6)*P[n - 1, x])/(36 + (126*n - 42)*x + (49*n^2 + 96*n - 72)*x^2 + (56*n^2 - 14*n - 14)*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] = ((216 + (756*n + 756)*x + (294*n^2 + 2046*n + 1098)*x^2 + (679*n^2 + 2198*n + 847)*x^3 + (586*n^2 + 1170*n + 366)*x^4 + (224*n^2 + 308*n + 84)*x^5 + (32*n^2 + 32*n + 8)*x^6)*v[i - 1])/(36 + (126*n - 42)*x + (49*n^2 + 96*n - 72)*x^2 + (56*n^2 - 14*n - 14)*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_A007318 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (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.