basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
only rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-4 - 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | -4 | -6 | -10 | |||||||||||||
| 2 | 16 | 48 | 36 | 100 | ||||||||||||
| 3 | -64 | -288 | -432 | -216 | -1000 | |||||||||||
| 4 | 256 | 1536 | 3456 | 3456 | 1296 | 10000 | ||||||||||
| 5 | -1024 | -7680 | -23040 | -34560 | -25920 | -7776 | -100000 | |||||||||
| 6 | 4096 | 36864 | 138240 | 276480 | 311040 | 186624 | 46656 | 1000000 | ||||||||
| 7 | -16384 | -172032 | -774144 | -1935360 | -2903040 | -2612736 | -1306368 | -279936 | -10000000 | |||||||
| 8 | 65536 | 786432 | 4128768 | 12386304 | 23224320 | 27869184 | 20901888 | 8957952 | 1679616 | 100000000 | ||||||
| 9 | -262144 | -3538944 | -21233664 | -74317824 | -167215104 | -250822656 | -250822656 | -161243136 | -60466176 | -10077696 | -1000000000 | |||||
| 10 | 1048576 | 15728640 | 106168320 | 424673280 | 1114767360 | 2006581248 | 2508226560 | 2149908480 | 1209323520 | 403107840 | 60466176 | 10000000000 | ||||
| 11 | -4194304 | -69206016 | -519045120 | -2335703040 | -7007109120 | -14714929152 | -22072393728 | -23648993280 | -17736744960 | -8868372480 | -2660511744 | -362797056 | -100000000000 | |||
| 12 | 16777216 | 301989888 | 2491416576 | 12457082880 | 42042654720 | 100902371328 | 176579149824 | 227030335488 | 212840939520 | 141893959680 | 63852281856 | 17414258688 | 2176782336 | 1000000000000 | ||
| 13 | -67108864 | -1308622848 | -11777605632 | -64776830976 | -242913116160 | -655865413632 | -1311730827264 | -1967596240896 | -2213545771008 | -1844621475840 | -1106772885504 | -452770725888 | -113192681472 | -13060694016 | -10000000000000 | |
| 14 | 268435456 | 5637144576 | 54962159616 | 329772957696 | 1360313450496 | 4080940351488 | 9182115790848 | 15740769927168 | 20659760529408 | 20659760529408 | 15494820397056 | 8451720216576 | 3169395081216 | 731398864896 | 78364164096 | 100000000000000 |
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Technical details
- Node id
n-ca6d44114ccba1b3· revision 0 · JSON- Definition source
derived:basis_map_A013609- Fingerprint
f6-f06e51c139cfae05
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, -6; 2: 16, 48, 36; 3: -64, -288, -432, -216; 4: 256, 1536, 3456, 3456, 1296; 5: -1024, -7680, -23040, -34560, -25920, -7776; 6: 4096, 36864, 138240, 276480, 311040, 186624, 46656; 7: -16384, -172032, -774144, -1935360, -2903040, -2612736, -1306368, -279936; 8: 65536, 786432, 4128768, 12386304, 23224320, 27869184, 20901888, 8957952, 1679616; 9: -262144, -3538944, -21233664, -74317824, -167215104, -250822656, -250822656, -161243136, -60466176, -10077696;
%F FormulaThe row polynomials satisfy P(n,x) = -(4+6*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 + 6*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 + 6*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]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
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 → x ↦ x+1 → basis A013610
- A038221 → basis A013611 → x ↦ x+1
- basis-map coefficients of A127647 → basis A013610
- A178821 → basis A013610 → D
- A156992 → basis A007318 → basis A013609
- A085880 → basis A013610 → D
- A038325 → basis A007318 → basis A013609
- A085880 → basis A013609 → basis A013610
- A027465 → basis A013611 → basis A013610
- A038221 → basis A013610 → D
- A038325 → basis A013610
- A178821 → basis A013610 → reverse
- A038220 → D
- A085881 → basis A013610
- A038334 → D
- A085881 → basis A013611 → basis A007318
- A038338 → basis A013610
- basis-map coefficients of A113214, shifted → basis A013610
- basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
- A178822 → basis A013611 → basis A007318
- A038338 → basis A013610 → D
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
- A038221 → basis A013609 → basis A013610
- A136572 → basis A013609 → x ↦ x+1
- A038325 → basis A013609 → basis A013610
- A134059 → basis A013610
- A038243 → basis A013611
- A178821 → basis A013610
- basis-map coefficients of A112227 → basis A013610 → basis A013609
- A135065 → x ↦ x+1 → basis A013609
- A178820 → basis A013611 → basis A007318
- A038334 → reverse
- A196347 → basis A013611 → x ↦ x+1
- A135089 → basis A013610
- A156992 → basis A013610 → D
- A094305 → basis A013610
- A135065 → basis A013610
- A376827 → basis A013610
- A094305 → x ↦ x+1 → basis A013609
- A038208 → basis A013610 → D
- A028326 → basis A013610 → D
- basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
- A013620: Triangle of coefficients in expansion of (2+3x)^n.
- A178820 → basis A013610
- A038330 → basis A013609 → x ↦ x/(x+1)
- A216973 → basis A013610
- A028326 → D → basis A013610
- A178820 → basis A013609 → basis A013610
- A120910 → D → basis A013609
- A120910 → basis A013610 → basis A013610
- A094305 → x ↦ x+1 → x ↦ x/(x+1)
- basis-map coefficients of A026729 → basis A007318 → basis A013610
- A085880 → basis A013610 → reverse
- A003506 → D → basis A013610
- A196347 → basis A013610 → D
- A131431 → x ↦ x+1 → basis A013610
- A134058 → basis A013610
- basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
- A120910 → basis A013609
- basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
- A356546 → basis A013611 → basis A007318
- A120909 → basis A013610 → x ↦ x+1
- A356546 → basis A013610
- A196347 → basis A013610
- A135065 → x ↦ x+1 → x ↦ x/(x+1)
- A085881 → basis A007318 → basis A013609
- A038208 → D → basis A013610
- A038243 → basis A013611 → D
- basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
- A178822 → x ↦ x+1 → basis A013609
- basis-map coefficients of A113214, shifted → D → basis A013610
- A028326 → basis A007318 → basis A013609
- basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
- A178820 → basis A007318 → basis A013609
- A038330 → basis A013609 → basis A013609
- A038291 → basis A013609 → basis A013611
- A120910 → basis A013609 → reverse
- A156992 → basis A013610
- A038221 → basis A013610
- A085880 → basis A013611 → x ↦ x+1
- A344565 → basis A013610
- A038330 → basis A013611 → basis A013610
- A028326 → basis A013611 → x ↦ x+1
- A038338 → basis A013609 → basis A013610
- A258758 → basis A013610
- A356546 → x ↦ x+1 → basis A013609
- A013609 → basis A013610 → x ↦ x+1
- A178822 → basis A013610
- A003506 → x ↦ x+1 → basis A013609
- A085881 → basis A013609 → basis A013610
- A178821 → basis A013611 → basis A007318
- A356546 → basis A013609 → basis A013610
- A178822 → basis A013609 → basis A013610
- A124860 → basis A013610
- A156992 → basis A013609 → basis A013610
- A085880 → D → basis A013610
- A136572 → basis A013610 → basis A013609
- A134346 → basis A013610
- A028326 → basis A013609 → basis A013610
- A178822 → basis A013610 → D
- A135838 → basis A013610
- A178820 → basis A013610 → D
- A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
- A132200 → basis A013610
- A120910 → basis A013609 → D
- A038328 → basis A013609 → basis A013610
- A027466 → basis A013611 → x ↦ x/(x+1)
- A178821 → basis A013609 → basis A013610
- A156992 → basis A013611 → x ↦ x+1
- A038208 → basis A013610
- A028326 → basis A013610
- A013620 → D
- A120910 → x ↦ x/(x+1)
- A013610 → basis A013611 → basis A013609
- A038338 → basis A013610 → reverse
- A013610 → D → basis A013609
- A085880 → basis A013610
- basis-map coefficients of A026729 → basis A013610 → 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.