basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).
palindromicγ-positivenonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 1 | 2 | |||||||||||||
| 2 | 2 | 4 | 2 | 8 | ||||||||||||
| 3 | 3 | 9 | 9 | 3 | 24 | |||||||||||
| 4 | 5 | 20 | 30 | 20 | 5 | 80 | ||||||||||
| 5 | 8 | 40 | 80 | 80 | 40 | 8 | 256 | |||||||||
| 6 | 13 | 78 | 195 | 260 | 195 | 78 | 13 | 832 | ||||||||
| 7 | 21 | 147 | 441 | 735 | 735 | 441 | 147 | 21 | 2688 | |||||||
| 8 | 34 | 272 | 952 | 1904 | 2380 | 1904 | 952 | 272 | 34 | 8704 | ||||||
| 9 | 55 | 495 | 1980 | 4620 | 6930 | 6930 | 4620 | 1980 | 495 | 55 | 28160 | |||||
| 10 | 89 | 890 | 4005 | 10680 | 18690 | 22428 | 18690 | 10680 | 4005 | 890 | 89 | 91136 | ||||
| 11 | 144 | 1584 | 7920 | 23760 | 47520 | 66528 | 66528 | 47520 | 23760 | 7920 | 1584 | 144 | 294912 | |||
| 12 | 233 | 2796 | 15378 | 51260 | 115335 | 184536 | 215292 | 184536 | 115335 | 51260 | 15378 | 2796 | 233 | 954368 | ||
| 13 | 377 | 4901 | 29406 | 107822 | 269555 | 485199 | 646932 | 646932 | 485199 | 269555 | 107822 | 29406 | 4901 | 377 | 3088384 | |
| 14 | 610 | 8540 | 55510 | 222040 | 610610 | 1221220 | 1831830 | 2093520 | 1831830 | 1221220 | 610610 | 222040 | 55510 | 8540 | 610 | 9994240 |
b-files: A016095, A016095 (reversed triangle), A063727 (row sums) ?
Compute more
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Technical details
- Node id
n-ed81f100961f7473· revision 0 · JSON- Definition source
derived:basis_map_coefficients_of:oeis_catalog:A127647- Fingerprint
f6-77a7ed3b1d5f257e
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: yes
- γ-positive (gamma_positive): yes
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes
Matches ?
- A016095 reversed triangle 55 terms, 10 rows · tabl · nonn b-file?
- A016095 triangle 55 terms, 10 rows · tabl · nonn b-file?
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 2, 4, 2; 3: 3, 9, 9, 3; 4: 5, 20, 30, 20, 5; 5: 8, 40, 80, 80, 40, 8; 6: 13, 78, 195, 260, 195, 78, 13; 7: 21, 147, 441, 735, 735, 441, 147, 21; 8: 34, 272, 952, 1904, 2380, 1904, 952, 272, 34; 9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;
%F FormulaThe row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (1 + 2*x + x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (1 + 2*x + x^2)*v[i - 2]); 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
A016095 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 2, 4, 2; 3: 3, 9, 9, 3; 4: 5, 20, 30, 20, 5; 5: 8, 40, 80, 80, 40, 8; 6: 13, 78, 195, 260, 195, 78, 13; 7: 21, 147, 441, 735, 735, 441, 147, 21; 8: 34, 272, 952, 1904, 2380, 1904, 952, 272, 34; 9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;
%F FormulaThe row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (1 + 2*x + x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (1 + 2*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A016095 reversed triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 2, 4, 2; 3: 3, 9, 9, 3; 4: 5, 20, 30, 20, 5; 5: 8, 40, 80, 80, 40, 8; 6: 13, 78, 195, 260, 195, 78, 13; 7: 21, 147, 441, 735, 735, 441, 147, 21; 8: 34, 272, 952, 1904, 2380, 1904, 952, 272, 34; 9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (1 + 2*x + x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (1 + 2*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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
derived from Row-hypergeometric rows (multiplier sequences)
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences)
Proof chains ? JSON
-
Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
Related ?
Same fingerprint ?
- A135089: Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.
- A178821 → basis A013609
- A134058: Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.
- A007318 → ×(−(1/2)n+(1/2)n²) → D
- A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
- basis-map coefficients of A021012 → x ↦ x+1
- A134059: Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.
- A376827: T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).
- A135838 → basis A013609
- veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.
- A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
- A027465 → basis A013611
- veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
- A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
- A258758: Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).
- A219570: Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled b…
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
- A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
- basis-map coefficients of A113214, shifted → D
- A120910 → basis A013610
- A038208 → D
- Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.
- basis-map coefficients of A026729, shifted → basis A007318
- A085880 → basis A013609
- A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
- A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
- A178822 → basis A013609
- A156992 → basis A013609
- A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
- A135838 → D
- veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.
- A178820 → basis A013609
- A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
- A344565 → basis A013609
- A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
- A196347 → basis A013609
- A344565 → D
- A380114: Triangle read by rows: The convolution triangle of 2^n, where the convolution triangle of a sequence is define…
- basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
- A127952: Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).
- A038330 → basis A013611
- A097805: Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triang…
- basis-map coefficients of A026729 → basis A007318
- Riordan array (1-x,x/(1-x)).
- A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
- A216973: Exponential Riordan array [x*exp(x),x].
- A038221 → basis A013609
- veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
- veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
- A131431 → x ↦ x+1
- A216973 → D
- A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
- A134346: Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).
- A007318: Binomial
- A178820 → D
- A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
- A085880 → D
- A038338 → basis A013609
- A178822 → D
- A135065: A127733 * A007318 as infinite lower triangular matrices.
- A003506: Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.
- A124860: A Jacobsthal-Pascal triangle.
- basis-map coefficients of A113214, shifted → basis A013609
- A135065 → D
- A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
- A094305 → D
- A085881 → basis A013609
- A132200: Numbers in (4,4)-Pascal triangle .
- A178821 → D
- A028326 → basis A013609
- A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
- basis-map coefficients of A112227 → x ↦ x+1
- A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A00010…
- A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
- A038325 → basis A013609
- A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
- A356546 → basis A013609
- A028326 → D
References
- [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.