A272866 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 10\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)

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

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	4	10	5																											19
2	16	80	140	100	25																									361
3	64	480	1440	2200	1800	750	125																							6859
4	256	2560	10880	25600	36400	32000	17000	5000	625																					130321
5	1024	12800	70400	224000	456000	620000	570000	350000	137500	31250	3125																			2476099
6	4096	61440	414720	1664000	4416000	8160000	10760000	10200000	6900000	3250000	1012500	187500	15625																	47045881
7	16384	286720	2293760	11110400	36377600	85120000	146720000	189200000	183400000	133000000	71050000	27125000	7000000	1093750	78125															893871739
8	65536	1310720	12124160	68812800	268083200	759808000	1619968000	2649600000	3359200000	3312000000	2531200000	1484000000	654500000	210000000	46250000	6250000	390625													16983563041
9	262144	5898240	61931520	403046400	1821081600	6064128000	15418368000	30597120000	48032640000	60088000000	60040800000	47808000000	30114000000	14805000000	5557500000	1537500000	295312500	35156250	1953125											322687697779
10	1048576	26214400	308019200	2260992000	11624448000	44482560000	131420160000	306892800000	575193600000	873664000000	1081206400000	1092080000000	898740000000	599400000000	320850000000	135750000000	44343750000	10781250000	1835937500	195312500	9765625									6131066257801
11	4194304	115343360	1499463680	12255232000	70647808000	305479680000	1028628480000	2764185600000	6026803200000	10781056000000	15937433600000	19548704000000	19921792000000	16845400000000	11771100000000	6748500000000	3139125000000	1165312500000	336875000000	73046875000	11171875000	1074218750	48828125							116490258898219
12	16777216	503316480	7172259840	64592281600	412640870400	1989672960000	7522549760000	22870425600000	56892211200000	117213184000000	201694310400000	291474432000000	354861376000000	364343040000000	315147360000000	228932000000000	138897000000000	69795000000000	28696250000000	9487500000000	2459531250000	481250000000	66796875000	5859375000	244140625					2213314919066161
13	67108864	2181038080	33806090240	332608307200	2332347596800	12408061952000	52050132992000	176655564800000	493885849600000	1152126976000000	2263370137600000	3768906752000000	5342661376000000	6463358080000000	6678326720000000	5888916800000000	4420644800000000	2812810000000000	1507220000000000	673887500000000	248194375000000	73957812500000	17377343750000	3097656250000	393554687500	31738281250	1220703125			42052983462257059
14	268435456	9395240960	157370286080	1679399321600	12824503910400	74618765312000	343942889472000	1289163898880000	4002349711360000	10430644224000000	23044179558400000	43469963264000000	70376563712000000	98124579840000000	118060194560000000	122655724800000000	109963380800000000	84902272000000000	56260204000000000	31831800000000000	15267752500000000	6147212500000000	2050059375000000	555953125000000	119437500000000	19550781250000	2290039062500	170898437500	6103515625	799006685782884121

Technical details

Node id: n-6494b967cfaa76e5 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-c64e29f1cee5cd50

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:      4,      10,        5;
  2:     16,      80,      140,       100,         25;
  3:     64,     480,     1440,      2200,       1800,        750,         125;
  4:    256,    2560,    10880,     25600,      36400,      32000,       17000,        5000,         625;
  5:   1024,   12800,    70400,    224000,     456000,     620000,      570000,      350000,      137500,       31250,        3125;
  6:   4096,   61440,   414720,   1664000,    4416000,    8160000,    10760000,    10200000,     6900000,     3250000,     1012500,      187500,       15625;
  7:  16384,  286720,  2293760,  11110400,   36377600,   85120000,   146720000,   189200000,   183400000,   133000000,    71050000,    27125000,     7000000,     1093750,      78125;
  8:  65536, 1310720, 12124160,  68812800,  268083200,  759808000,  1619968000,  2649600000,  3359200000,  3312000000,  2531200000,  1484000000,   654500000,   210000000,   46250000,    6250000,    390625;
  9: 262144, 5898240, 61931520, 403046400, 1821081600, 6064128000, 15418368000, 30597120000, 48032640000, 60088000000, 60040800000, 47808000000, 30114000000, 14805000000, 5557500000, 1537500000, 295312500, 35156250, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (4+10*x+5*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[(4 + 10*x + 5*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] = (4 + 10*x + 5*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272866 → basis A013609

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.