{"id":"n-314823098ef7e229","revision":0,"first_index":1,"fingerprint":"f6-65341f67bf7e02f3","name":"reversal of Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["1","4","3"],["1","9","21","13"],["1","16","78","136","73"],["1","25","210","730","1045","501"],["1","36","465","2720","7515","9276","4051"],["1","49","903","8015","36575","85071","93289","37633"],["1","64","1596","20048","137270","519456","1053724","1047376","394353"],["1","81","2628","44436","427518","2404206","7836276","14196708","12975561","4596553"],["1","100","4095","89760","1156890","9083592","43632750","125685120","206844885","175721140","58941091"],["1","121","6105","168465","2805330","29379042","195911562","825594330","2140967565","3241760005","2581284541","824073141"],["1","144","8778","297880","6229575","84028032","744371628","4332365136","16330034655","38658650800","54388827306","40864292184","12470162233"],["1","169","12246","501358","12870715","217530027","2474768868","19063622292","98778836871","337931065615","738247378726","972672654174","693347907421","202976401213"],["1","196","16653","809536","25038013","518570052","7376265897","72774008736","497374664787","2329110699916","7314900744151","14874602354976","18470852510383","12548540320876","3535017524403"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","8","44","304","2512","24064","261536","3173888","42483968","621159424","9841950208","167879268352","3065723549696","59651093528576"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-3","2","6"],["0","2","-3","-5"],["0","0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2","4","6"],["0","0","0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","1","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181573,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-cb9470d644e9e0f9"]},{"id":181574,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-cb9470d644e9e0f9"]},{"id":181575,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-cb9470d644e9e0f9"]}],"region_exact":{},"incoming_edges":[{"id":"e-cb9470d644e9e0f9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-314823098ef7e229","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":"ccfdd067b39b93cf3ca62e4850c36db9e19e5414b550f0832508651cfafc90d9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-01897e65d9bb3867","role":"sequence","name":"Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-a3f094afbd76129b","kind":"fingerprint_related","node_a_id":"n-01897e65d9bb3867","node_b_id":"n-314823098ef7e229","fingerprint":"f6-65341f67bf7e02f3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-65341f67bf7e02f3"},"other_id":"n-01897e65d9bb3867","other":{"id":"n-01897e65d9bb3867","name":"Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","first_index":1,"fingerprint":"f6-65341f67bf7e02f3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["3","4","1"]],"card_rows":[["1"],["1","1"],["3","4","1"],["13","21","9","1"]],"row_sums_preview":["1","2","8","44","304","2512","24064","261536"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A059110"],"derivation":{"root":"n-01897e65d9bb3867","steps":[],"edges":[],"text":"A059110"},"headline":"A059110: Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  1;\n   3: 1,  4,    3;\n   4: 1,  9,   21,    13;\n   5: 1, 16,   78,   136,     73;\n   6: 1, 25,  210,   730,   1045,     501;\n   7: 1, 36,  465,  2720,   7515,    9276,    4051;\n   8: 1, 49,  903,  8015,  36575,   85071,   93289,    37633;\n   9: 1, 64, 1596, 20048, 137270,  519456, 1053724,  1047376,   394353;\n  10: 1, 81, 2628, 44436, 427518, 2404206, 7836276, 14196708, 12975561, 4596553;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n-3)*x+(n^2-3*n+2)*x^2+(n^2-5*n+6)*x^3)*P(n-1,x) - (2*x^2+(2*n-4)*x^3+(2*n-6)*x^4)*d/dx P(n-1,x) + (x^4+x^5)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n - 3)*x + (n^2 - 3*n + 2)*x^2 + (n^2 - 5*n + 6)*x^3)*P[n - 1, x] - (2*x^2 + (2*n - 4)*x^3 + (2*n - 6)*x^4)*D[P[n - 1, x], x] + (x^4 + x^5)*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (2*n - 3)*x + (n^2 - 3*n + 2)*x^2 + (n^2 - 5*n + 6)*x^3)*v[i - 1] - (2*x^2 + (2*n - 4)*x^3 + (2*n - 6)*x^4)*deriv(v[i - 1], x) + (x^4 + x^5)*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-01897e65d9bb3867","steps":["reverse"],"edges":["e-cb9470d644e9e0f9"],"text":"A059110 \u2192 reverse"},"headline":"A059110 \u2192 reverse","chain":null}
