A035607 → basis A013609 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(4 + 15\,x + 11\,x^{2} - 8\,x^{3} - 4\,x^{4} - 2\,n - 12\,n x - 24\,n x^{2} - n x^{3} + 2\,n x^{4} - 2\,n^{2} x + 8\,n^{2} x^{2} + 2\,n^{2} x^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-4 - 15\,x - 11\,x^{2} + 8\,x^{3} + 4\,x^{4} + 2\,n + 20\,n x + 26\,n x^{2} - 9\,n x^{3} - 6\,n x^{4} + 8\,n^{2} + 4\,n^{2} x - 20\,n^{2} x^{2} - 6\,n^{2} x^{3} - 4\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x + 38\,x^{2} + 87\,x^{3} + 85\,x^{4} + 32\,x^{5} + 4\,x^{6} + 4\,n + 32\,n x + 86\,n x^{2} + 92\,n x^{3} + 34\,n x^{4} + 4\,n x^{5} + 4\,n^{2} + 16\,n^{2} x + 18\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||||||||
| 1 | 1 | 1 | |||||||||||||
| 2 | 7 | 4 | 11 | ||||||||||||
| 3 | 23 | 32 | 8 | 63 | |||||||||||
| 4 | 73 | 168 | 104 | 16 | 361 | ||||||||||
| 5 | 217 | 704 | 744 | 288 | 32 | 1985 | |||||||||
| 6 | 623 | 2620 | 3976 | 2640 | 736 | 64 | 10659 | ||||||||
| 7 | 1743 | 9024 | 17968 | 17280 | 8256 | 1792 | 128 | 56191 | |||||||
| 8 | 4785 | 29456 | 72816 | 92512 | 64192 | 23808 | 4224 | 256 | 292049 | ||||||
| 9 | 12945 | 92416 | 273328 | 433472 | 398720 | 214784 | 64896 | 9728 | 512 | 1500801 | |||||
| 10 | 34615 | 281268 | 969328 | 1847136 | 2127744 | 1521536 | 667008 | 169728 | 22016 | 1024 | 7641403 | ||||
| 11 | 91687 | 835680 | 3290232 | 7331200 | 10176192 | 9129216 | 5313920 | 1958912 | 430080 | 49152 | 2048 | 38608319 | |||
| 12 | 240953 | 2434872 | 10785816 | 27534320 | 44799552 | 48516864 | 35458432 | 17349376 | 5509120 | 1062912 | 108544 | 4096 | 193804857 | ||
| 13 | 629033 | 6980544 | 34368600 | 98947424 | 184840224 | 235037952 | 207584384 | 127557120 | 53722624 | 14966784 | 2574336 | 237568 | 8192 | 967454785 | |
| 14 | 1632863 | 19742060 | 106966200 | 342970160 | 723876320 | 1058441664 | 1098826880 | 816405760 | 431761920 | 159400960 | 39528448 | 6131712 | 516096 | 16384 | 4806217427 |
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Technical details
- Node id
n-712d4076e029103f· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-38a0582a03ef8145
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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 ExampleTriangle begins: 0: 1; 1: 1; 2: 7, 4; 3: 23, 32, 8; 4: 73, 168, 104, 16; 5: 217, 704, 744, 288, 32; 6: 623, 2620, 3976, 2640, 736, 64; 7: 1743, 9024, 17968, 17280, 8256, 1792, 128; 8: 4785, 29456, 72816, 92512, 64192, 23808, 4224, 256; 9: 12945, 92416, 273328, 433472, 398720, 214784, 64896, 9728, 512;
%F FormulaThe row polynomials satisfy -(2*n-4-(2*n^3-2*n^2-12*n+15)*x-(8*n^2-24*n+11)*x^2-(2*n^2-n-8)*x^3-(2*n-4)*x^4)*P(n,x) = -(4*n^3-8*n^2-2*n+4+(6*n^3-4*n^2-20*n+15)*x+(20*n^2-26*n+11)*x^2+(6*n^2+9*n-8)*x^3+(6*n-4)*x^4)*P(n-1,x) + (4*n^2+4*n+(16*n^2+32*n+6)*x+(18*n^2+86*n+38)*x^2+(4*n^2+92*n+87)*x^3+(34*n+85)*x^4+(4*n+32)*x^5+4*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 8*n^2 - 2*n + 4 + (6*n^3 - 4*n^2 - 20*n + 15)*x + (20*n^2 - 26*n + 11)*x^2 + (6*n^2 + 9*n - 8)*x^3 + (6*n - 4)*x^4)*P[n - 1, x] + (4*n^2 + 4*n + (16*n^2 + 32*n + 6)*x + (18*n^2 + 86*n + 38)*x^2 + (4*n^2 + 92*n + 87)*x^3 + (34*n + 85)*x^4 + (4*n + 32)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(-2*n + 4 + (2*n^3 - 2*n^2 - 12*n + 15)*x + (8*n^2 - 24*n + 11)*x^2 + (2*n^2 - n - 8)*x^3 + (2*n - 4)*x^4)]];
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*n^3 - 8*n^2 - 2*n + 4 + (6*n^3 - 4*n^2 - 20*n + 15)*x + (20*n^2 - 26*n + 11)*x^2 + (6*n^2 + 9*n - 8)*x^3 + (6*n - 4)*x^4)*v[i - 1] + (4*n^2 + 4*n + (16*n^2 + 32*n + 6)*x + (18*n^2 + 86*n + 38)*x^2 + (4*n^2 + 92*n + 87)*x^3 + (34*n + 85)*x^4 + (4*n + 32)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(-2*n + 4 + (2*n^3 - 2*n^2 - 12*n + 15)*x + (8*n^2 - 24*n + 11)*x^2 + (2*n^2 - n - 8)*x^3 + (2*n - 4)*x^4)); 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 Laguerre’s theorem on differential operators
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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
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
basis_map_A013609 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
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Geometric basis map (elementary) polytool ? step ›
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from
Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
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from
basis_map_A013609 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
References
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.