A139548 → basis A013609 → basis A013609

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 64\,x + 32\,x^{2}\right)\,P_{n-1}(x)\)

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

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	32	64	32																											128
2	1024	4096	6144	4096	1024																									16384
3	32768	196608	491520	655360	491520	196608	32768																							2097152
4	1048576	8388608	29360128	58720256	73400320	58720256	29360128	8388608	1048576																					268435456
5	33554432	335544320	1509949440	4026531840	7046430720	8455716864	7046430720	4026531840	1509949440	335544320	33554432																			34359738368
6	1073741824	12884901888	70866960384	236223201280	531502202880	850403524608	992137445376	850403524608	531502202880	236223201280	70866960384	12884901888	1073741824																	4398046511104
7	34359738368	481036337152	3126736191488	12506944765952	34394098106368	68788196212736	103182294319104	117922622078976	103182294319104	68788196212736	34394098106368	12506944765952	3126736191488	481036337152	34359738368															562949953421312
8	1099511627776	17592186044416	131941395333120	615726511554560	2001111162552320	4802666790125568	8804889115230208	12578413021757440	14150714649477120	12578413021757440	8804889115230208	4802666790125568	2001111162552320	615726511554560	131941395333120	17592186044416	1099511627776													72057594037927936
9	35184372088832	633318697598976	5383208929591296	28710447624486912	107664178591825920	301459700057112576	653162683457077248	1119707457354989568	1539597753863110656	1710664170959011840	1539597753863110656	1119707457354989568	653162683457077248	301459700057112576	107664178591825920	28710447624486912	5383208929591296	633318697598976	35184372088832											9223372036854775808
10	1125899906842624	22517998136852480	213920982300098560	1283525893800591360	5454985048652513280	17455952155688042496	43639880389220106240	87279760778440212480	141829611264965345280	189106148353287127040	208016763188615839744	189106148353287127040	141829611264965345280	87279760778440212480	43639880389220106240	17455952155688042496	5454985048652513280	1283525893800591360	213920982300098560	22517998136852480	1125899906842624									1180591620717411303424
11	36028797018963968	792633534417207296	8322652111380676608	55484347409204510720	263550650193721425920	948782340697397133312	2688216631975958544384	6144495158802190958592	11520928422754108047360	17921444213173056962560	23297877477124974051328	25415866338681789874176	23297877477124974051328	17921444213173056962560	11520928422754108047360	6144495158802190958592	2688216631975958544384	948782340697397133312	263550650193721425920	55484347409204510720	8322652111380676608	792633534417207296	36028797018963968							151115727451828646838272
12	1152921504606846976	27670116110564327424	318206335271489765376	2333513125324258279424	12250943907952355966976	49003775631809423867904	155178622834063175581696	399030744430448165781504	847940331914702352285696	1507449478959470848507904	2261174218439206272761856	2877858096195353438060544	3117679604211632891232256	2877858096195353438060544	2261174218439206272761856	1507449478959470848507904	847940331914702352285696	399030744430448165781504	155178622834063175581696	49003775631809423867904	12250943907952355966976	2333513125324258279424	318206335271489765376	27670116110564327424	1152921504606846976					19342813113834066795298816
13	36893488147419103232	959230691832896684032	11990383647911208550400	95923069183289668403200	551557647803915593318400	2426853650337228610600960	8493987776180300137103360	24268536503372286106009600	57637774195509179501772800	115275548391018359003545600	195968432264731210306027520	285044992385063578626949120	356306240481329473283686400	383714412826047125074739200	356306240481329473283686400	285044992385063578626949120	195968432264731210306027520	115275548391018359003545600	57637774195509179501772800	24268536503372286106009600	8493987776180300137103360	2426853650337228610600960	551557647803915593318400	95923069183289668403200	11990383647911208550400	959230691832896684032	36893488147419103232			2475880078570760549798248448
14	1180591620717411303424	33056565380087516495872	446263632631181472694272	3867618149470239430017024	24172613434188996437606400	116028544484107182900510720	444776087189077534451957760	1397867702594243679706152960	3669402719309889659228651520	8154228265133088131619225600	15493033703752867450076528640	25352236969777419463761592320	35915669040518010906995589120	44203900357560628808609955840	47361321811672102294939238400	44203900357560628808609955840	35915669040518010906995589120	25352236969777419463761592320	15493033703752867450076528640	8154228265133088131619225600	3669402719309889659228651520	1397867702594243679706152960	444776087189077534451957760	116028544484107182900510720	24172613434188996437606400	3867618149470239430017024	446263632631181472694272	33056565380087516495872	1180591620717411303424	316912650057057350374175801344

Technical details

Node id: n-d311e24793b27ad5 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-2cd9f46ca2d53115

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: yes
γ-positive (gamma_positive): yes
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:              1;
  1:             32,              64,               32;
  2:           1024,            4096,             6144,              4096,               1024;
  3:          32768,          196608,           491520,            655360,             491520,             196608,              32768;
  4:        1048576,         8388608,         29360128,          58720256,           73400320,           58720256,           29360128,             8388608,             1048576;
  5:       33554432,       335544320,       1509949440,        4026531840,         7046430720,         8455716864,         7046430720,          4026531840,          1509949440,           335544320,            33554432;
  6:     1073741824,     12884901888,      70866960384,      236223201280,       531502202880,       850403524608,       992137445376,        850403524608,        531502202880,        236223201280,         70866960384,         12884901888,         1073741824;
  7:    34359738368,    481036337152,    3126736191488,    12506944765952,     34394098106368,     68788196212736,    103182294319104,     117922622078976,     103182294319104,      68788196212736,      34394098106368,      12506944765952,      3126736191488,       481036337152,        34359738368;
  8:  1099511627776,  17592186044416,  131941395333120,   615726511554560,   2001111162552320,   4802666790125568,   8804889115230208,   12578413021757440,   14150714649477120,   12578413021757440,    8804889115230208,    4802666790125568,   2001111162552320,    615726511554560,    131941395333120,    17592186044416,    1099511627776;
  9: 35184372088832, 633318697598976, 5383208929591296, 28710447624486912, 107664178591825920, 301459700057112576, 653162683457077248, 1119707457354989568, 1539597753863110656, 1710664170959011840, 1539597753863110656, 1119707457354989568, 653162683457077248, 301459700057112576, 107664178591825920, 28710447624486912, 5383208929591296, 633318697598976, 35184372088832;

%F Formula

The row polynomials satisfy P(n,x) = (32+64*x+32*x^2)*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[(32 + 64*x + 32*x^2)*P[n - 1, x]];
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] = (32 + 64*x + 32*x^2)*v[i - 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (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) 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.