{"id":"n-39cdbb608deef22e","revision":0,"first_index":0,"fingerprint":"f6-93af6d1da209b590","name":"reversal of basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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"],["6","4"],["27","48","16"],["108","360","288","64"],["405","2160","3024","1536","256"],["1458","11340","24192","20736","7680","1024"],["5103","54432","163296","207360","126720","36864","4096"],["17496","244944","979776","1710720","1520640","718848","172032","16384"],["59049","1049760","5388768","12317184","14826240","10063872","3870720","786432","65536"],["196830","4330260","27713664","80061696","124540416","113218560","61931520","20054016","3538944","262144"],["649539","17321040","135104112","480370176","934053120","1086898176","789626880","360972288","100859904","15728640","1048576"],["2125764","67552056","630485856","2702082240","6404935680","9238634496","8527970304","5143855104","2017198080","495452160","69206016","4194304"],["6908733","257926032","2837186352","14411105280","40831464960","71269466112","81015717888","61726261248","31770869760","10899947520","2387607552","301989888","16777216"],["22320522","967222620","12380449536","73496636928","244988789760","507794946048","694420439040","648125743104","419375480832","188024094720","57302581248","11324620800","1308622848","67108864"],["71744535","3571283520","52616910528","360801672192","1396436101632","3385299640320","5468560957440","6110899863552","4822818029568","2707546963968","1074423398400","294440140800","52999225344","5637144576","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","91","820","7381","66430","597871","5380840","48427561","435848050","3922632451","35303692060","317733228541","2859599056870","25736391511831"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["3","2"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184668,"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-378f55faea7baf1a"]},{"id":184669,"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-378f55faea7baf1a"]},{"id":184670,"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-378f55faea7baf1a"]}],"region_exact":{},"incoming_edges":[{"id":"e-378f55faea7baf1a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-39cdbb608deef22e","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"a65675a32559fe1b4daf1c76b17fdc1d0a3361f325a33a9740d3e68be62ac66c","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-08fe62d9c34c34e9","role":"sequence","name":"basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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-0606008b06be1783","kind":"fingerprint_related","node_a_id":"n-39cdbb608deef22e","node_b_id":"n-6d0fd56cd80060c6","fingerprint":"f6-93af6d1da209b590","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-93af6d1da209b590"},"other_id":"n-6d0fd56cd80060c6","other":{"id":"n-6d0fd56cd80060c6","name":"basis_map_A013610 of veronese_section_2_1 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-93af6d1da209b590","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["8","12"]],"card_rows":[["2"],["8","12"],["32","96","54"],["128","576","720","216"]],"row_sums_preview":["0","2","20","182","1640","14762","132860","1195742"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-7b8f3a8fe3cbfab2","e-6aca3823e63cb3df"],"text":"A034870 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A034870 \u2192 Veronese 2 (j=1) \u2192 basis A013610","chain":null}},{"id":"l-e930c6d834e8f77f","kind":"fingerprint_related","node_a_id":"n-39cdbb608deef22e","node_b_id":"n-06f760567d49aa7f","fingerprint":"f6-93af6d1da209b590","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-93af6d1da209b590"},"other_id":"n-06f760567d49aa7f","other":{"id":"n-06f760567d49aa7f","name":"derivative of basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","first_index":1,"fingerprint":"f6-93af6d1da209b590","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["9"],["60","90"]],"card_rows":[["9"],["60","90"],["336","1008","567"],["1728","7776","9720","2916"]],"row_sums_preview":["0","9","150","1911","22140","243573","2590770","26904195"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["basis A013610","D"],"edges":["e-4aa0aeb00b1923c9","e-c87b991a75c18873"],"text":"A091042 \u2192 basis A013610 \u2192 D"},"headline":"A091042 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-f0857809ba61c31e","kind":"fingerprint_related","node_a_id":"n-08fe62d9c34c34e9","node_b_id":"n-39cdbb608deef22e","fingerprint":"f6-93af6d1da209b590","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-93af6d1da209b590"},"other_id":"n-08fe62d9c34c34e9","other":{"id":"n-08fe62d9c34c34e9","name":"basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","first_index":0,"fingerprint":"f6-93af6d1da209b590","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","6"],["16","48","27"]],"card_rows":[["1"],["4","6"],["16","48","27"],["64","288","360","108"]],"row_sums_preview":["1","10","91","820","7381","66430","597871","5380840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2245b35ec7cd4857","steps":["basis A013610"],"edges":["e-0cc36dd83eba3acf"],"text":"A091044 \u2192 basis A013610"},"headline":"A091044 \u2192 basis A013610","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:      6,       4;\n  2:     27,      48,       16;\n  3:    108,     360,      288,       64;\n  4:    405,    2160,     3024,     1536,       256;\n  5:   1458,   11340,    24192,    20736,      7680,      1024;\n  6:   5103,   54432,   163296,   207360,    126720,     36864,     4096;\n  7:  17496,  244944,   979776,  1710720,   1520640,    718848,   172032,    16384;\n  8:  59049, 1049760,  5388768, 12317184,  14826240,  10063872,  3870720,   786432,   65536;\n  9: 196830, 4330260, 27713664, 80061696, 124540416, 113218560, 61931520, 20054016, 3538944, 262144;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (3*n+3+(2*n+2)*x)*P(n-1,x) + (6*x+2*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[((3*n + 3 + (2*n + 2)*x)*P[n - 1, x] + (6*x + 2*x^2)*D[P[n - 1, x], x])/(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] = ((3*n + 3 + (2*n + 2)*x)*v[i - 1] + (6*x + 2*x^2)*deriv(v[i - 1], x))/(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-2245b35ec7cd4857","steps":["basis A013610","reverse"],"edges":["e-0cc36dd83eba3acf","e-378f55faea7baf1a"],"text":"A091044 \u2192 basis A013610 \u2192 reverse"},"headline":"A091044 \u2192 basis A013610 \u2192 reverse","chain":null}
