{"id":"n-1ebae075a566c573","revision":0,"first_index":0,"fingerprint":"f6-ade2b5b96bd8d678","name":"reversal of basis_map_A013609 of Even row Pascal-square read by antidiagonals.","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"],["4","3"],["12","20","8"],["32","88","76","21"],["80","320","444","260","55"],["192","1040","2048","1896","840","144"],["448","3136","8176","10592","7344","2616","377"],["1024","8960","29568","50080","48160","26664","7940","987"],["2304","24576","99456","210816","262960","200704","92436","23644","2584"],["5120","65280","316416","814080","1261760","1242320","785280","309512","69380","6765"],["11264","168960","963072","2939904","5494720","6689088","5432560","2928864","1008708","201220","17711"],["24576","428032","2827264","10061568","22178816","32410112","32425344","22391648","10520576","3216912","578064","46368"],["53248","1064960","8053760","32947200","84198400","144501760","172950400","146707840","88074400","36660800","10078560","1647600","121393"],["114688","2609152","22364160","103966720","303846400","602184960","843786240","854004480","628346880","333524800","124596160","31111440","4664836","317811"],["245760","6307840","60755968","317890560","1050649600","2372531200","3827692800","4524241920","3963321600","2573491200","1223927200","414668800","94839132","13132580","832040"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","40","217","1159","6160","32689","173383","919480","4875913","25856071","137109280","727060321","3855438727","20444528200"],"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"],["4"]]},"terms":[{"coeff":{"coeffs":[["8","14","5"],["8","-2","-5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","16","18","5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180085,"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-6d67f02ade321477"]},{"id":180086,"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-6d67f02ade321477"]},{"id":180087,"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-6d67f02ade321477"]}],"region_exact":{},"incoming_edges":[{"id":"e-6d67f02ade321477","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-1ebae075a566c573","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":"01e787554027e0b53a1355e65c26b095ef254d4cac2ccbbad720d0a057b2bb67","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-7c7a913d07444057","role":"sequence","name":"basis_map_A013609 of Even row Pascal-square read by antidiagonals.","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-0562b215899be40f","kind":"fingerprint_related","node_a_id":"n-1ebae075a566c573","node_b_id":"n-2de3c606f08cd985","fingerprint":"f6-ade2b5b96bd8d678","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ade2b5b96bd8d678"},"other_id":"n-2de3c606f08cd985","other":{"id":"n-2de3c606f08cd985","name":"basis_map_A013609 of basis_map_A013611 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","first_index":0,"fingerprint":"f6-ade2b5b96bd8d678","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","16"],["128","320","192"]],"card_rows":[["1"],["12","16"],["128","320","192"],["1344","4864","5632","2048"]],"row_sums_preview":["1","28","640","13888","296704","6307840","133894144","2840707072"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2245b35ec7cd4857","steps":["basis A013611","basis A013609"],"edges":["e-61d2480a3633cd97","e-558cf966b6b39f46"],"text":"A091044 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A091044 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-43a3c30074eb13b7","kind":"fingerprint_related","node_a_id":"n-7c7a913d07444057","node_b_id":"n-1ebae075a566c573","fingerprint":"f6-ade2b5b96bd8d678","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ade2b5b96bd8d678"},"other_id":"n-7c7a913d07444057","other":{"id":"n-7c7a913d07444057","name":"basis_map_A013609 of Even row Pascal-square read by antidiagonals.","first_index":0,"fingerprint":"f6-ade2b5b96bd8d678","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","4"],["8","20","12"]],"card_rows":[["1"],["3","4"],["8","20","12"],["21","76","88","32"]],"row_sums_preview":["1","7","40","217","1159","6160","32689","173383"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79ebafb266da3b41","steps":["basis A013609"],"edges":["e-1a56a7670c2d3c41"],"text":"A172431 \u2192 basis A013609"},"headline":"A172431 \u2192 basis A013609","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:    4,     3;\n  2:   12,    20,      8;\n  3:   32,    88,     76,     21;\n  4:   80,   320,    444,    260,      55;\n  5:  192,  1040,   2048,   1896,     840,     144;\n  6:  448,  3136,   8176,  10592,    7344,    2616,    377;\n  7: 1024,  8960,  29568,  50080,   48160,   26664,   7940,    987;\n  8: 2304, 24576,  99456, 210816,  262960,  200704,  92436,  23644,  2584;\n  9: 5120, 65280, 316416, 814080, 1261760, 1242320, 785280, 309512, 69380, 6765;\n","recurrence_sentence":"The row polynomials satisfy 4*n*P(n,x) = (8*n+8-(2*n-14)*x-(5*n-5)*x^2)*P(n-1,x) + (16*x+18*x^2+5*x^3)*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[((8*n + 8 - (2*n - 14)*x - (5*n - 5)*x^2)*P[n - 1, x] + (16*x + 18*x^2 + 5*x^3)*D[P[n - 1, x], x])/(4*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] = ((8*n + 8 - (2*n - 14)*x - (5*n - 5)*x^2)*v[i - 1] + (16*x + 18*x^2 + 5*x^3)*deriv(v[i - 1], x))/(4*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-79ebafb266da3b41","steps":["basis A013609","reverse"],"edges":["e-1a56a7670c2d3c41","e-6d67f02ade321477"],"text":"A172431 \u2192 basis A013609 \u2192 reverse"},"headline":"A172431 \u2192 basis A013609 \u2192 reverse","chain":null}