{"id":"n-807761fed4e5ed8c","revision":0,"first_index":0,"fingerprint":"f6-6004798a4a421655","name":"shift_x_plus_one of basis_map_A013610 of Pascal-(1,3,1) array.","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":false,"rows":[["1"],["5","3"],["37","39","9"],["245","387","189","27"],["1669","3471","2565","783","81"],["11285","29211","28989","13635","2997","243"],["76453","236583","294597","187623","63909","10935","729"],["517685","1864371","2793501","2246427","1041093","276291","38637","2187"],["3505861","14401983","25225029","24515487","14400261","5211135","1129221","133407","6561"],["23741525","109559115","219632445","250351155","178258725","81941787","24229773","4428675","452709","19683"],["160777957","823396695","1859157765","2431864215","2035789605","1136621079","427425093","106596567","16828965","1515591","59049"],["1088788085","6127700451","15387600285","22717700235","21866805765","14378104179","6575222061","2086470171","449225109","62375427","5019165","177147"],["7373275909","45232628463","125041566213","205666092495","223803382725","169452497583","91369113669","35284001679","9666852741","1829200239","226571013","16474671","531441"],["49931836565","331614468219","1000706223933","1814555966499","2202876162405","1888011138123","1173508663437","534741990003","178355201589","42938090523","7242064605","809384643","53675541","1594323"],["338138493733","2417018875527","7905966975045","15668265710151","20994684136677","20096344593351","14153078573829","7439965983495","2929560225957","858708134919","184206248901","28009952199","2851180965","173781207","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","85","848","8569","86360","870829","8780192","88528945","892615592","9000032581","90745188080","914962188841","9225346460408","93016977648925"],"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"],["9"]]},"terms":[{"coeff":{"coeffs":[["135","54"],["-126","-27"],["36","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["615","603","108"],["-219","-198","-27"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["292","483","234","27"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182255,"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-f8bb333f0139bb49"]},{"id":182256,"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-f8bb333f0139bb49"]},{"id":182257,"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-f8bb333f0139bb49"]}],"region_exact":{},"incoming_edges":[{"id":"e-f8bb333f0139bb49","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-807761fed4e5ed8c","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":"7778b5b1d17a5f4776f732b87996fd3bd0986f062dc3aae14b812b18eabb1d27","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-9813839be9bee901","role":"sequence","name":"basis_map_A013610 of Pascal-(1,3,1) array.","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-ca8677a065ade55d","kind":"candidate_equal","node_a_id":"n-807761fed4e5ed8c","node_b_id":"n-9deb9a70f73e3627","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-9deb9a70f73e3627","other":{"id":"n-9deb9a70f73e3627","name":"basis_map_A007318 of basis_map_A013610 of Pascal-(1,3,1) array.","first_index":0,"fingerprint":"f6-6004798a4a421655","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","3"],["37","39","9"]],"card_rows":[["1"],["5","3"],["37","39","9"],["245","387","189","27"]],"row_sums_preview":["1","8","85","848","8569","86360","870829","8780192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d8f9ad9ce0e039d0","steps":["basis A013610","basis A007318"],"edges":["e-a15ab36c8834192d","e-c505fbd8cbcb986c"],"text":"A081578 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A081578 \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:        5,         3;\n  2:       37,        39,         9;\n  3:      245,       387,       189,        27;\n  4:     1669,      3471,      2565,       783,        81;\n  5:    11285,     29211,     28989,     13635,      2997,      243;\n  6:    76453,    236583,    294597,    187623,     63909,    10935,      729;\n  7:   517685,   1864371,   2793501,   2246427,   1041093,   276291,    38637,    2187;\n  8:  3505861,  14401983,  25225029,  24515487,  14400261,  5211135,  1129221,  133407,   6561;\n  9: 23741525, 109559115, 219632445, 250351155, 178258725, 81941787, 24229773, 4428675, 452709, 19683;\n","recurrence_sentence":"The row polynomials satisfy 9*n*P(n,x) = (36*n^2-126*n+135-(27*n-54)*x)*P(n-1,x) - (219*n-615+(198*n-603)*x+(27*n-108)*x^2)*d/dx P(n-1,x) + (292+483*x+234*x^2+27*x^3)*d^2/dx^2 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[((36*n^2 - 126*n + 135 - (27*n - 54)*x)*P[n - 1, x] - (219*n - 615 + (198*n - 603)*x + (27*n - 108)*x^2)*D[P[n - 1, x], x] + (292 + 483*x + 234*x^2 + 27*x^3)*D[P[n - 1, x], {x, 2}])/(9*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] = ((36*n^2 - 126*n + 135 - (27*n - 54)*x)*v[i - 1] - (219*n - 615 + (198*n - 603)*x + (27*n - 108)*x^2)*deriv(v[i - 1], x) + (292 + 483*x + 234*x^2 + 27*x^3)*deriv(deriv(v[i - 1], x), x))/(9*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-d8f9ad9ce0e039d0","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-a15ab36c8834192d","e-f8bb333f0139bb49"],"text":"A081578 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A081578 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}
