{"id":"n-0b73360ed752ea1a","revision":0,"first_index":0,"fingerprint":"f6-2b1205fb0a5fe514","name":"reversal of Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.","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"],["1","1"],["1","6","1"],["1","21","27","1"],["1","58","270","112","1"],["1","141","1738","2878","453","1"],["1","318","8739","39320","28167","1818","1"],["1","685","37665","375755","769955","264411","7279","1"],["1","1434","146560","2858960","13243650","13905746","2430652","29124","1"],["1","2949","531456","18617280","169140810","414525726","239506500","22108860","116505","1"],["1","5998","1833237","108482304","1757950770","8576223516","12039688650","4008974880","200028285","466030","1"],["1","12117","6099511","582421413","15732621306","137596928826","393021867126","333064015290","65943852645","1804914865","1864131","1"],["1","24378","19764690","2939636752","125839240983","1832469336540","9493402582356","16813299338856","8920095056055","1073150790970","16264739226","7456536","1"],["1","48925","62804502","14150074498","923159880115","21187794538863","183731855615052","599683121246292","685559631703311","233733884311075","17349324787006","146472131466","29826157","1"],["1","98046","196681831","65644146280","6327958414833","219442755244678","3002334346953267","16556303729333664","35554509626398107","27017485584429946","6034189680433941","279346862169688","1318636923235","119304642","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","8","50","442","5212","78364","1455752","32616128","864550088","26693653672","947754597232","38277487967344","1742329486967264","88671325140532160"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"],["0","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184130,"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-bbb984223fa70a80"]},{"id":184131,"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-bbb984223fa70a80"]},{"id":184132,"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-bbb984223fa70a80"]}],"region_exact":{},"incoming_edges":[{"id":"e-bbb984223fa70a80","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-0b73360ed752ea1a","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":"93b40842a364488fc9f442f8dddba2cddb09c30c8538f929b9a14424ade7b4b3","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-0c97d42b0cd05e48","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.","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-52fd9b30f5e2dd59","kind":"fingerprint_related","node_a_id":"n-0c97d42b0cd05e48","node_b_id":"n-0b73360ed752ea1a","fingerprint":"f6-2b1205fb0a5fe514","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2b1205fb0a5fe514"},"other_id":"n-0c97d42b0cd05e48","other":{"id":"n-0c97d42b0cd05e48","name":"Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.","first_index":0,"fingerprint":"f6-2b1205fb0a5fe514","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","6","1"]],"card_rows":[["1"],["1","1"],["1","6","1"],["1","27","21","1"]],"row_sums_preview":["1","2","8","50","442","5212","78364","1455752"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A166960"],"derivation":{"root":"n-0c97d42b0cd05e48","steps":[],"edges":[],"text":"A166960"},"headline":"A166960: Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 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: 1;\n  1: 1,    1;\n  2: 1,    6,      1;\n  3: 1,   21,     27,        1;\n  4: 1,   58,    270,      112,         1;\n  5: 1,  141,   1738,     2878,       453,         1;\n  6: 1,  318,   8739,    39320,     28167,      1818,         1;\n  7: 1,  685,  37665,   375755,    769955,    264411,      7279,        1;\n  8: 1, 1434, 146560,  2858960,  13243650,  13905746,   2430652,    29124,      1;\n  9: 1, 2949, 531456, 18617280, 169140810, 414525726, 239506500, 22108860, 116505, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+n^2*x)*P(n-1,x) + (x-(2*n-1)*x^2)*d/dx P(n-1,x) + 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[(1 + n^2*x)*P[n - 1, x] + (x - (2*n - 1)*x^2)*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];\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] = (1 + n^2*x)*v[i - 1] + (x - (2*n - 1)*x^2)*deriv(v[i - 1], x) + x^3*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-0c97d42b0cd05e48","steps":["reverse"],"edges":["e-bbb984223fa70a80"],"text":"A166960 \u2192 reverse"},"headline":"A166960 \u2192 reverse","chain":null}
