basis-map coefficients of A026729, shifted → basis A013610 → basis A013610
only rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-4 - 9\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | -4 | -9 | -13 | |||||||||||||
| 3 | 16 | 72 | 81 | 169 | ||||||||||||
| 4 | -64 | -432 | -972 | -729 | -2197 | |||||||||||
| 5 | 256 | 2304 | 7776 | 11664 | 6561 | 28561 | ||||||||||
| 6 | -1024 | -11520 | -51840 | -116640 | -131220 | -59049 | -371293 | |||||||||
| 7 | 4096 | 55296 | 311040 | 933120 | 1574640 | 1417176 | 531441 | 4826809 | ||||||||
| 8 | -16384 | -258048 | -1741824 | -6531840 | -14696640 | -19840464 | -14880348 | -4782969 | -62748517 | |||||||
| 9 | 65536 | 1179648 | 9289728 | 41803776 | 117573120 | 211631616 | 238085568 | 153055008 | 43046721 | 815730721 | ||||||
| 10 | -262144 | -5308416 | -47775744 | -250822656 | -846526464 | -1904684544 | -2857026816 | -2754990144 | -1549681956 | -387420489 | -10604499373 | |||||
| 11 | 1048576 | 23592960 | 238878720 | 1433272320 | 5643509760 | 15237476352 | 28570268160 | 36733201920 | 30993639120 | 15496819560 | 3486784401 | 137858491849 | ||||
| 12 | -4194304 | -103809024 | -1167851520 | -7882997760 | -35473489920 | -111741493248 | -251418359808 | -404065221120 | -454573373760 | -340930030320 | -153418513644 | -31381059609 | -1792160394037 | |||
| 13 | 16777216 | 452984832 | 5605687296 | 42042654720 | 212840939520 | 766227382272 | 2011346878464 | 3879026122752 | 5454880485120 | 5454880485120 | 3682044327456 | 1506290861232 | 282429536481 | 23298085122481 | ||
| 14 | -67108864 | -1962934272 | -26499612672 | -218621804544 | -1229747650560 | -4980477984768 | -14941433954304 | -33618226397184 | -56730757045248 | -70913446306560 | -63822101675904 | -39163562392032 | -14686335897012 | -2541865828329 | -302875106592253 | |
| 15 | 268435456 | 8455716864 | 123664859136 | 1112983732224 | 6886586843136 | 30989640794112 | 104590037680128 | 268945811177472 | 529487065755648 | 794230598633472 | 893509423462656 | 731053164651264 | 411217405116336 | 142344486386424 | 22876792454961 | 3937376385699289 |
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Technical details
- Node id
n-e22cf82a694daa01· 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 -1)
- 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: -4, -9; 3: 16, 72, 81; 4: -64, -432, -972, -729; 5: 256, 2304, 7776, 11664, 6561; 6: -1024, -11520, -51840, -116640, -131220, -59049; 7: 4096, 55296, 311040, 933120, 1574640, 1417176, 531441; 8: -16384, -258048, -1741824, -6531840, -14696640, -19840464, -14880348, -4782969; 9: 65536, 1179648, 9289728, 41803776, 117573120, 211631616, 238085568, 153055008, 43046721; 10: -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 >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4 + 9*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); 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, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: 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 ≥ 1, 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 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
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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 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
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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 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
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from
basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
Related ?
Same fingerprint ?
- A013610 → basis A013610
- A038328 → basis A013609 → basis A013609
- A038279 → basis A013611
- A038271 → basis A013610 → x ↦ x/(x+1)
- A013610 → basis A013611 → basis A013610
- A013610 → D → basis A013610
- A013610 → basis A013610 → D
- A038334 → basis A013609 → basis A007318
- A013622 → basis A013609 → x ↦ x/(x+1)
- A130123 → basis A013610 → basis A013610
- A038329 → basis A013611 → x ↦ x+1
- A136572 → basis A013610 → basis A013610
- basis-map coefficients of A026729 → basis A013610 → basis A013610
- A038319 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A030528, shifted → basis A013610 → basis A013610
- basis-map coefficients of A112227 → basis A013610 → basis A013610
- A038279 → basis A013611 → D