A200536 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 14\,n x + 8\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 14 + 16\,x\)

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	P_n(1)
0																													0
1	14	16																											30
2	168	584	672	256																									1680
3	1512	8784	20328	23424	13440	3072																							70560
4	12096	98496	342720	660544	761600	525312	200704	32768																					2634240
5	90720	950400	4415040	11936640	20698720	23873280	18314240	9011200	2580480	327680																			92198400
6	653184	8366976	48625920	169240320	391950720	634217856	731641344	601743360	345784320	132218880	30277632	3145728																	3097866240
7	4572288	69237504	483138432	2056997376	5961755520	12420963072	19138587776	22081712128	19077617664	12189614080	5598838784	1750597632	333971456	29360128															101196963840
8	31352832	547928064	4462553088	22468368384	78209832960	199367239680	384478918656	571197949952	659106717696	590717747200	407849795584	213033418752	81489035264	21550333952	3523215360	268435456													3238302842880
9	211631616	4192321536	39034275840	226878050304	921956152320	2779749052416	6439155222528	11705434288128	16905648324096	19509057146880	17988751097856	13178069581824	7575975493632	3346217828352	1096600780800	251255586816	35936796672	2415919104											102006539550720
10	1410877440	31240857600	327323566080	2156985262080	10022559805440	34889894277120	94363408097280	202909745848320	352025790842880	497085437470720	573671659151360	541092655595520	415398706544640	257295022817280	126704953589760	48477430087680	13895561379840	2808505958400	357019156480	21474836480									3173536786022400
11	9311791104	227917172736	2653860464640	19551805194240	102246915133440	403602193539072	1248301391597568	3099431876935680	6278578296883200	10487566829752320	14539175646865408	16780106927603712	16122768959897600	12856902600622080	8454104662671360	4535403524653056	1953266720047104	659129169346560	167888258334720	30354681364480	3472481058816	188978561024							97744933009489920
12	60949905408	1634037940224	20932906401792	170496234602496	991022887895040	4373982997807104	15227142960218112	42874503591591936	99327193299468288	191586493254942720	310134126406680576	423414147556098048	488696199186284544	476837494323412992	392403726615183360	271009207887593472	155834276691050496	73727713411596288	28212296379531264	8515196792340480	1951534355054592	319184789569536	33191507263488	1649267441664					2985663408289873920
13	396174385152	11545653510144	161375032885248	1439869600825344	9208926851604480	44939037318709248	173904036403249152	547537127556317184	1427858633240911872	3123353668992860160	5782435977938780160	9116309151158796288	12286135628169568256	14180925346247016448	14018026613184921600	11845607989069217792	8524061415451918336	5191463135348785152	2652147311198601216	1122208790473605120	386338693280956416	105471344384147456	21970011849490432	3280392941469696	312673619148800	14293651161088			90565123384792842240
14	2559896027136	80575774949376	1220217106268160	11836282008305664	82602946229207040	441622630229114880	1880568475363049472	6547037079189454848	18979588790033743872	46426627584378961920	96755546784033865728	172995617792349241344	266651139686347014144	355408225933520764928	410232522594380021760	410063686618901905408	354444561956326080512	264115925813355872256	168846309784589172736	91963867888751738880	42265067078666944512	16174566257259970560	5060599857085939712	1261112761585238016	240758686744576000	33068361961177088	2909307767095296	123145302310912	2730886797449138012160

Technical details

Node id: n-3fbcf027e0e0a748 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-cc36510735305ef0

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         0;
  1:        14,         16;
  2:       168,        584,         672,          256;
  3:      1512,       8784,       20328,        23424,        13440,          3072;
  4:     12096,      98496,      342720,       660544,       761600,        525312,        200704,          32768;
  5:     90720,     950400,     4415040,     11936640,     20698720,      23873280,      18314240,        9011200,        2580480,         327680;
  6:    653184,    8366976,    48625920,    169240320,    391950720,     634217856,     731641344,      601743360,      345784320,      132218880,       30277632,        3145728;
  7:   4572288,   69237504,   483138432,   2056997376,   5961755520,   12420963072,   19138587776,    22081712128,    19077617664,    12189614080,     5598838784,     1750597632,     333971456,      29360128;
  8:  31352832,  547928064,  4462553088,  22468368384,  78209832960,  199367239680,  384478918656,   571197949952,   659106717696,   590717747200,   407849795584,   213033418752,   81489035264,   21550333952,    3523215360,    268435456;
  9: 211631616, 4192321536, 39034275840, 226878050304, 921956152320, 2779749052416, 6439155222528, 11705434288128, 16905648324096, 19509057146880, 17988751097856, 13178069581824, 7575975493632, 3346217828352, 1096600780800, 251255586816, 35936796672, 2415919104;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (6*n+14*n*x+8*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 14 + 16*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 14 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 14*n*x + 8*n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 14 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 14*n*x + 8*n*x^2)*v[i - 1])/(n - 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 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

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.