basis-map coefficients of A030528, shifted → basis A013610 → basis A013610
only rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-4 - 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | -4 | -9 | -13 | ||||||
| 2 | 16 | 72 | 81 | 169 | |||||
| 3 | -64 | -432 | -972 | -729 | -2197 | ||||
| 4 | 256 | 2304 | 7776 | 11664 | 6561 | 28561 | |||
| 5 | -1024 | -11520 | -51840 | -116640 | -131220 | -59049 | -371293 | ||
| 6 | 4096 | 55296 | 311040 | 933120 | 1574640 | 1417176 | 531441 | 4826809 | |
| 7 | -16384 | -258048 | -1741824 | -6531840 | -14696640 | -19840464 | -14880348 | -4782969 | -62748517 |
Compute more
?
Technical details
- Node id
n-3ac4aedfd42ceede· revision 0 · JSON- Definition source
derived:basis_map_A013610- Fingerprint
f6-12e1fe8c90bb4450
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): no - 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 ExampleTriangle begins: 0: 1; 1: -4, -9; 2: 16, 72, 81; 3: -64, -432, -972, -729; 4: 256, 2304, 7776, 11664, 6561; 5: -1024, -11520, -51840, -116640, -131220, -59049; 6: 4096, 55296, 311040, 933120, 1574640, 1417176, 531441; 7: -16384, -258048, -1741824, -6531840, -14696640, -19840464, -14880348, -4782969; 8: 65536, 1179648, 9289728, 41803776, 117573120, 211631616, 238085568, 153055008, 43046721; 9: -262144, -5308416, -47775744, -250822656, -846526464, -1904684544, -2857026816, -2754990144, -1549681956, -387420489;
%F FormulaThe row polynomials satisfy P(n,x) = -(4+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4 + 9*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(4 + 9*x)*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]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A013610 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
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Geometric basis map (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
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from
basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
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from
basis_map_A013610 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
Related ?
Same fingerprint ?
- basis-map coefficients of A112227 → basis A013610 → basis A013610
- A013610 → basis A013610
- A038328 → basis A013609 → basis A013609
- A013622 → basis A013609 → x ↦ x/(x+1)
- A038334 → basis A013609 → basis A007318
- A038329 → basis A013611 → x ↦ x+1
- A038319 → basis A013611 → x ↦ x/(x+1)
- A013610 → D → basis A013610
- A136572 → basis A013610 → basis A013610
- A038271 → basis A013610 → x ↦ x/(x+1)
- A013610 → basis A013611 → basis A013610
- A038279 → basis A013611 → D
- A038279 → basis A013611
- A130123 → basis A013610 → basis A013610
- A013610 → basis A013610 → D
- basis-map coefficients of A026729 → basis A013610 → basis A013610
- basis-map coefficients of A026729, shifted → basis A013610 → basis A013610
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.