{"id":"n-8f297cd827c43952","revision":1,"first_index":0,"fingerprint":"f6-f25c978562657419","name":"basis_map_A013610 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1","3"],["2","9","9"],["2","15","36","27"],["4","36","117","162","81"],["4","48","225","513","567","243"],["8","108","594","1701","2673","2187","729"],["8","132","918","3483","7776","10206","7290","2187"],["16","288","2232","9720","26001","43740","45198","26244","6561"],["16","336","3096","16416","55161","121743","176418","161838","85293","19683"],["32","720","7200","42120","159570","408969","718065","852930","656100","295245","59049"],["32","816","9360","63720","285930","887679","1944972","3007125","3214890","2263545","944784","177147"],["64","1728","21168","155520","763020","2633148","6552981","11849166","15451155","14171760","8680203","3188646","531441"],["64","1920","26352","219024","1229580","4922208","14452425","31508109","50998653","60525225","51195483","29229255","10097379","1594323"],["128","4032","58464","517104","3116232","13533156","43671474","106373493","196521633","274046409","283966641","212044959","107882523","33480783","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","20","80","400","1600","8000","32000","160000","640000","3200000","12800000","64000000","256000000","1280000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","9","9"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166941,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-473a0c8b41a1e681"]},{"id":166942,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-473a0c8b41a1e681"]},{"id":166943,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-473a0c8b41a1e681"]}],"region_exact":{},"incoming_edges":[{"id":"e-473a0c8b41a1e681","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-8f297cd827c43952","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"cd3f4843c3e88dfaf2f27ef12c4cdf26023f16b5eb74d98035b30a44ba51e772","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-35bd700768cac55f","role":"sequence","name":"Triangle read by rows: triangle of repeated stepped binomial coefficients.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  1,   3;\n  2:  2,   9,    9;\n  3:  2,  15,   36,    27;\n  4:  4,  36,  117,   162,    81;\n  5:  4,  48,  225,   513,   567,    243;\n  6:  8, 108,  594,  1701,  2673,   2187,    729;\n  7:  8, 132,  918,  3483,  7776,  10206,   7290,   2187;\n  8: 16, 288, 2232,  9720, 26001,  43740,  45198,  26244,  6561;\n  9: 16, 336, 3096, 16416, 55161, 121743, 176418, 161838, 85293, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+9*x+9*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 9*x + 9*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 9*x + 9*x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-35bd700768cac55f","steps":["basis A013610"],"edges":["e-473a0c8b41a1e681"],"text":"A103633 \u2192 basis A013610"},"headline":"A103633 \u2192 basis A013610","chain":null}