{"id":"n-7ed1ee07114ce3f2","revision":0,"first_index":0,"fingerprint":"f6-bdd2ae59e2c72e1b","name":"shift_x_plus_one of basis_map_A013611 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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"],["7","4"],["54","60","16"],["413","684","368","64"],["3161","6956","5632","1984","256"],["24192","66408","71824","38208","9984","1024"],["185149","609048","824384","587200","231936","48128","4096"],["1417003","5432740","8831632","7886272","4175104","1309696","225280","16384"],["10844766","47483028","90110496","96763968","64279296","27036672","7028736","1032192","65536"],["82998377","408591972","886594704","1112547648","889431552","469622784","163713024","36323328","4653056","262144"],["635212469","3472931516","8481015408","12178474176","11384663808","7237386240","3167772672","942391296","182321152","20709376","1048576"],["4861479168","29226323856","79326175328","128282497920","137303891712","102106199040","53821009920","20104298496","5214371840","894173184","91226112","4194304"],["37206416521","243935691120","728485325824","1310098619264","1579894449408","1345684546560","829960273920","373397176320","121598967808","27949531136","4303355904","398458880","16777216"],["284752311487","2021953039876","6588676217312","13043348829056","17499305073152","16799116185600","11869989949440","6249426862080","2451270533120","707370942464","145954439168","20388511744","1728053248","67108864"],["2179298263014","16661184646764","58826715074288","127122581072256","187808397552384","200639034122240","159818933944320","96412774809600","44258184724480","15382822256640","3984479748096","745743056896","95328141312","7449083904","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","130","1529","17989","211640","2489941","29294111","344644690","4054738589","47703926689","561235840880","6602929589881","77683348056611","913943194931650"],"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"],["16"]]},"terms":[{"coeff":{"coeffs":[["224","128"],["-160","-64"],["48","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["816","960","256"],["-276","-288","-64"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["345","636","368","64"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183177,"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-9685bd72c94b1d3d"]},{"id":183178,"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-9685bd72c94b1d3d"]},{"id":183179,"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-9685bd72c94b1d3d"]}],"region_exact":{},"incoming_edges":[{"id":"e-9685bd72c94b1d3d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7ed1ee07114ce3f2","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":"89c298f240ec9eaecaee761ae78c12d4b2686da248b898b07e11ec1a8b8b326c","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-44fd873021acce78","role":"sequence","name":"basis_map_A013611 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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-c02ebd73afe2b8cb","kind":"candidate_equal","node_a_id":"n-7ed1ee07114ce3f2","node_b_id":"n-21551e8b0efd7b64","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-21551e8b0efd7b64","other":{"id":"n-21551e8b0efd7b64","name":"basis_map_A007318 of basis_map_A013611 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","first_index":0,"fingerprint":"f6-bdd2ae59e2c72e1b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["7","4"],["54","60","16"]],"card_rows":[["1"],["7","4"],["54","60","16"],["413","684","368","64"]],"row_sums_preview":["1","11","130","1529","17989","211640","2489941","29294111"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d5c14b9fec875ecc","steps":["basis A013611","basis A007318"],"edges":["e-3c5bbf704ea0dfe8","e-51003081ad1f0db6"],"text":"A121574 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A121574 \u2192 basis A013611 \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:        7,         4;\n  2:       54,        60,        16;\n  3:      413,       684,       368,         64;\n  4:     3161,      6956,      5632,       1984,       256;\n  5:    24192,     66408,     71824,      38208,      9984,      1024;\n  6:   185149,    609048,    824384,     587200,    231936,     48128,      4096;\n  7:  1417003,   5432740,   8831632,    7886272,   4175104,   1309696,    225280,    16384;\n  8: 10844766,  47483028,  90110496,   96763968,  64279296,  27036672,   7028736,  1032192,   65536;\n  9: 82998377, 408591972, 886594704, 1112547648, 889431552, 469622784, 163713024, 36323328, 4653056, 262144;\n","recurrence_sentence":"The row polynomials satisfy 16*n*P(n,x) = (48*n^2-160*n+224-(64*n-128)*x)*P(n-1,x) - (276*n-816+(288*n-960)*x+(64*n-256)*x^2)*d/dx P(n-1,x) + (345+636*x+368*x^2+64*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[((48*n^2 - 160*n + 224 - (64*n - 128)*x)*P[n - 1, x] - (276*n - 816 + (288*n - 960)*x + (64*n - 256)*x^2)*D[P[n - 1, x], x] + (345 + 636*x + 368*x^2 + 64*x^3)*D[P[n - 1, x], {x, 2}])/(16*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] = ((48*n^2 - 160*n + 224 - (64*n - 128)*x)*v[i - 1] - (276*n - 816 + (288*n - 960)*x + (64*n - 256)*x^2)*deriv(v[i - 1], x) + (345 + 636*x + 368*x^2 + 64*x^3)*deriv(deriv(v[i - 1], x), x))/(16*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-d5c14b9fec875ecc","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-3c5bbf704ea0dfe8","e-9685bd72c94b1d3d"],"text":"A121574 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A121574 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}
