{"id":"n-7444a01f7c96b41d","revision":0,"first_index":0,"fingerprint":"f6-9162f2f6dcf54bcb","name":"shift_x_plus_one of basis_map_A013610 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["13","9"],["25","18"],["181","243","81"],["481","684","243"],["2653","5229","3402","729"],["8425","17766","12474","2916"],["40261","104391","100359","42282","6561"],["141361","393408","409941","189540","32805"],["624493","2008449","2553768","1600155","492075","59049"],["2320825","8001594","11013732","7564104","2591595","354294"],["9814741","37723419","59734989","49749876","22897890","5491557","531441"],["37664641","154629972","263914119","239642712","122073966","33067440","3720087"],["155441533","695643669","1320244758","1374256125","844597530","305047134","59521392","4782969"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","22","43","505","1408","12013","41581","293854","1167055","7337989","31846144","185943913","854712937","4759535110"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["3"]]},"terms":[{"coeff":{"coeffs":[["-72","-54"],["75","54"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-196","-291","-108"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175549,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-9df6c77d36f9498c"]},{"id":175550,"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-9df6c77d36f9498c"]},{"id":175551,"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-9df6c77d36f9498c"]}],"region_exact":{},"incoming_edges":[{"id":"e-9df6c77d36f9498c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7444a01f7c96b41d","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"118fb389d590ed93e09a8101a7c95feca29394068fdca5fea753dca2c0d51a30","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-8d5fed5e270db201","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","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":[{"id":"l-c9aa298549b77997","kind":"candidate_equal","node_a_id":"n-7444a01f7c96b41d","node_b_id":"n-29f3572152dfc897","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-29f3572152dfc897","other":{"id":"n-29f3572152dfc897","name":"basis_map_A007318 of basis_map_A013610 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","first_index":0,"fingerprint":"f6-9162f2f6dcf54bcb","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["13","9"]],"card_rows":[["1"],["1"],["13","9"],["25","18"]],"row_sums_preview":["1","1","22","43","505","1408","12013","41581"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e189038e95d51606","steps":["basis A013610","basis A007318"],"edges":["e-99e9156f8de29382","e-3a807118bc7c94f8"],"text":"A304236 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A304236 \u2192 basis A013610 \u2192 basis A007318","chain":null}}],"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;\n  2:     13,      9;\n  3:     25,     18;\n  4:    181,    243,     81;\n  5:    481,    684,    243;\n  6:   2653,   5229,   3402,    729;\n  7:   8425,  17766,  12474,   2916;\n  8:  40261, 104391, 100359,  42282,  6561;\n  9: 141361, 393408, 409941, 189540, 32805;\n","recurrence_sentence":"The row polynomials satisfy 3*n*P(n,x) = (75*n-72+(54*n-54)*x)*P(n-1,x) - (196+291*x+108*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((75*n - 72 + (54*n - 54)*x)*P[n - 1, x] - (196 + 291*x + 108*x^2)*D[P[n - 1, x], x])/(3*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((75*n - 72 + (54*n - 54)*x)*v[i - 1] - (196 + 291*x + 108*x^2)*deriv(v[i - 1], x))/(3*n)); 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-e189038e95d51606","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-99e9156f8de29382","e-9df6c77d36f9498c"],"text":"A304236 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A304236 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}