{"id":"n-5ad3fee9d3289d52","revision":0,"first_index":0,"fingerprint":"f6-a7854389020f6acb","name":"reversal of basis_map_A007318 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["2","7","6"],["4","20","33","18"],["4","28","73","84","36"],["8","68","230","387","324","108"],["8","84","366","847","1098","756","216"],["16","192","984","2792","4737","4806","2700","648"],["16","224","1368","4760","10321","14280","12312","6048","1296"],["32","496","3408","13624","34922","59523","67464","49032","20736","3888"],["32","560","4400","20440","62170","129367","186510","183960","118800","45360","7776"],["64","1216","10480","54080","185660","445244","761121","927450","789480","447120","151632","23328"],["64","1344","12912","75040","293820","816564","1651609","2449692","2644380","2026080","1045872","326592","46656"],["128","2880","29856","188816","812760","2514588","5752910","9854211","12637836","11985300","8169984","3790800","1073088","139968"],["128","3136","35616","248528","1190392","4140108","10782086","21360031","32346258","37260972","32140584","20130768","8654688","2286144","279936"],["256","6656","80640","603904","3126368","11851392","33984496","75066320","128772609","171560718","176064084","136683288","77701680","30536352","7418304","839808"],["256","7168","93952","765184","4334176","18104128","57687280","143035312","278905249","429105936","519185520","488811456","351068256","185939712","68491008","15676416","1679616"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["15","75","225","1125","3375","16875","50625","253125","759375","3796875","11390625","56953125","170859375","854296875","2562890625"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","7","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-5","-16","-12"],["-7","-19","-12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","18","26","12"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178962,"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-93a3897d23ed5968"]},{"id":178963,"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-93a3897d23ed5968"]},{"id":178964,"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-93a3897d23ed5968"]}],"region_exact":{},"incoming_edges":[{"id":"e-93a3897d23ed5968","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-5ad3fee9d3289d52","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":"e7772adac07f435111e17fedd2afdad87b5823d80827a88768ca2cafb8bfdfa7","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-73875e607efb6f6a","role":"sequence","name":"basis_map_A007318 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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-b4ce50affba0a326","kind":"candidate_equal","node_a_id":"n-22aa0691270f8663","node_b_id":"n-5ad3fee9d3289d52","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-22aa0691270f8663","other":{"id":"n-22aa0691270f8663","name":"mobius_x_over_x_plus_one of reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","first_index":0,"fingerprint":"f6-a7854389020f6acb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2","7","6"],["4","20","33","18"],["4","28","73","84","36"]],"card_rows":[["2","7","6"],["4","20","33","18"],["4","28","73","84","36"],["8","68","230","387","324","108"]],"row_sums_preview":["15","75","225","1125","3375","16875","50625","253125"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5a70a6e97131d437","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-75e40465d2024023","e-2ebc405f97fe63ca"],"text":"A064861 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A064861 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-233bc74e2ca62c17","kind":"fingerprint_related","node_a_id":"n-17ae8aea8bf71051","node_b_id":"n-5ad3fee9d3289d52","fingerprint":"f6-a7854389020f6acb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a7854389020f6acb"},"other_id":"n-17ae8aea8bf71051","other":{"id":"n-17ae8aea8bf71051","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","first_index":0,"fingerprint":"f6-a7854389020f6acb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["12","14","4"]],"card_rows":[["1"],["3","2"],["12","14","4"],["36","66","40","8"]],"row_sums_preview":["1","5","30","150","900","4500","27000","135000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-35bd700768cac55f","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-69752228c1d1e91e","e-07bfa876924909f5"],"text":"A103633 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A103633 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-ef60f3860e6c9150","kind":"fingerprint_related","node_a_id":"n-73875e607efb6f6a","node_b_id":"n-5ad3fee9d3289d52","fingerprint":"f6-a7854389020f6acb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a7854389020f6acb"},"other_id":"n-73875e607efb6f6a","other":{"id":"n-73875e607efb6f6a","name":"basis_map_A007318 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","first_index":0,"fingerprint":"f6-a7854389020f6acb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["6","7","2"],["18","33","20","4"],["36","84","73","28","4"]],"card_rows":[["6","7","2"],["18","33","20","4"],["36","84","73","28","4"],["108","324","387","230","68","8"]],"row_sums_preview":["15","75","225","1125","3375","16875","50625","253125"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5a70a6e97131d437","steps":["x \u21a6 x+1"],"edges":["e-4e787ca2d31dd33d"],"text":"A064861 \u2192 x \u21a6 x+1"},"headline":"A064861 \u2192 x \u21a6 x+1","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:  2,    7,     6;\n  1:  4,   20,    33,    18;\n  2:  4,   28,    73,    84,     36;\n  3:  8,   68,   230,   387,    324,    108;\n  4:  8,   84,   366,   847,   1098,    756,    216;\n  5: 16,  192,   984,  2792,   4737,   4806,   2700,    648;\n  6: 16,  224,  1368,  4760,  10321,  14280,  12312,   6048,   1296;\n  7: 32,  496,  3408, 13624,  34922,  59523,  67464,  49032,  20736,   3888;\n  8: 32,  560,  4400, 20440,  62170, 129367, 186510, 183960, 118800,  45360,   7776;\n  9: 64, 1216, 10480, 54080, 185660, 445244, 761121, 927450, 789480, 447120, 151632, 23328;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(7*n+5+(19*n+16)*x+(12*n+12)*x^2)*P(n-1,x) + (4+18*x+26*x^2+12*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 7*x + 6*x^2.","mathematica":"P[0, x_] := 2 + 7*x + 6*x^2;\nP[n_, x_] := P[n, x] = Expand[-(7*n + 5 + (19*n + 16)*x + (12*n + 12)*x^2)*P[n - 1, x] + (4 + 18*x + 26*x^2 + 12*x^3)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + 7*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = -(7*n + 5 + (19*n + 16)*x + (12*n + 12)*x^2)*v[i - 1] + (4 + 18*x + 26*x^2 + 12*x^3)*deriv(v[i - 1], x)); 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-5a70a6e97131d437","steps":["x \u21a6 x+1","reverse"],"edges":["e-4e787ca2d31dd33d","e-93a3897d23ed5968"],"text":"A064861 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A064861 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}
