A038328 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 + 140\,x + 64\,x^{2} + 84\,n x + 64\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(686 + 3528\,x + 3808\,x^{2} + 1152\,x^{3} + 1176\,n x + 2016\,n x^{2} + 768\,n x^{3} + 224\,n^{2} x^{2} + 128\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 196 + 896\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	196	896	576															1668
1	2744	18816	24192	8192														53944
2	38416	351232	677376	458752	102400													1628176
3	537824	6146560	15805440	16056320	7168000	1179648												46893792
4	7529536	103262208	331914240	449576960	301056000	99090432	12845056											1305274432
5	105413504	1686616064	6505519104	11014635520	9834496000	4855431168	1258815488	134217728										35395144576
6	1475789056	26985857024	121436356608	246727835648	275365888000	181269430272	70493667328	15032385536	1358954496									940146163968
7	20661046784	425027248128	2185854418944	5181284548608	6939220377600	5709987053568	2960734027776	947040288768	171228266496	13421772800								24554459049472
8	289254654976	6611534970880	38252452331520	103625690972160	161915142144000	159879637499904	103625690972160	44195213475840	11985978654720	1879048192000	129922760704							632389566628864
9	4049565169664	101817638551552	654541962117120	1994794551214080	3562133127168000	4103577362497536	3191671281942528	1701515718819840	615280237608960	144686710784000	20008105148416	1236950581248						16095313211602944
10	56693912375296	1555033025150976	10996304963567616	37236164955996160	74804795670528000	98485856699940864	89366795894390784	57170928152346624	25841769979576320	8102455803904000	1680680832466944	207807697649664	11613591568384					405516901179461632
11	793714773254144	23584667548123136	181938863942664192	677698202199130112	1512719201337344000	2240553239923654656	2323536693254160384	1734184820621180928	940640427256578048	368661739077632000	101961303836327936	18910500486119424	2113673665445888	107752139522048				10127404800061136896
12	11112006825558016	355584218417856512	2971668111063515136	12075349784639045632	29649296346211942400	48794270558337368064	56926648984726929408	48557174977393065984	30727587290381549568	14451540171843174400	4996103887980068864	1235486031759802368	207140019213697024	21119419346321408	989560464998400			250981071368604893184
13	155568095557812224	5333763276267847680	48003869486410629120	211318621231183298560	566032021154955264000	1024679681725084729344	1328288476310295019520	1274625843156567982080	921827618711446487040	505803906014511104000	209836363295162892288	64863016667389624320	14499801344958791680	2217539031363747840	207807697649664000	9007199254740992		6177702904398059634688
14	2177953337809371136	79650864925599858688	768061911782570065920	3641182396598850682880	10565931061559164928000	20866204427856270852096	29753661869350608437248	31724020985230136442880	25811173323920501637120	16185724992464355328000	7833890896352747978752	2905863146699055169536	811988875317692334080	165576247675159838720	23274462136762368000	2017612633061982208	81346268269379584	151140482374108616654848

Technical details

Node id: n-d04d98855e40597e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-18b5ed15f2b69ef1

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:           196,             896,             576;
  1:          2744,           18816,           24192,             8192;
  2:         38416,          351232,          677376,           458752,           102400;
  3:        537824,         6146560,        15805440,         16056320,          7168000,          1179648;
  4:       7529536,       103262208,       331914240,        449576960,        301056000,         99090432,         12845056;
  5:     105413504,      1686616064,      6505519104,      11014635520,       9834496000,       4855431168,       1258815488,        134217728;
  6:    1475789056,     26985857024,    121436356608,     246727835648,     275365888000,     181269430272,      70493667328,      15032385536,      1358954496;
  7:   20661046784,    425027248128,   2185854418944,    5181284548608,    6939220377600,    5709987053568,    2960734027776,     947040288768,    171228266496,     13421772800;
  8:  289254654976,   6611534970880,  38252452331520,  103625690972160,  161915142144000,  159879637499904,  103625690972160,   44195213475840,  11985978654720,   1879048192000,   129922760704;
  9: 4049565169664, 101817638551552, 654541962117120, 1994794551214080, 3562133127168000, 4103577362497536, 3191671281942528, 1701515718819840, 615280237608960, 144686710784000, 20008105148416, 1236950581248;

%F Formula

The row polynomials satisfy (49+(84*n+140)*x+(16*n^2+64*n+64)*x^2)*P(n,x) = (686+(1176*n+3528)*x+(224*n^2+2016*n+3808)*x^2+(128*n^2+768*n+1152)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 896*x + 576*x^2.

%t Mathematica

P[0, x_] := 196 + 896*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((686 + (1176*n + 3528)*x + (224*n^2 + 2016*n + 3808)*x^2 + (128*n^2 + 768*n + 1152)*x^3)*P[n - 1, x])/(49 + (84*n + 140)*x + (16*n^2 + 64*n + 64)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 896*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((686 + (1176*n + 3528)*x + (224*n^2 + 2016*n + 3808)*x^2 + (128*n^2 + 768*n + 1152)*x^3)*v[i - 1])/(49 + (84*n + 140)*x + (16*n^2 + 64*n + 64)*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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

Related ?

Same fingerprint ?

A038255 → basis A013611 → (θ+1)²

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.