{"id":"n-0ac9d6c345261ad6","revision":0,"first_index":0,"fingerprint":"f6-4600f723d5b687a0","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Riordan array (1, 2+2x).","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"],["2","4"],["6","22","20"],["16","88","160","96"],["44","320","868","1040","464"],["120","1088","3928","7056","6304","2240"],["328","3560","16040","38392","51472","36640","10816"],["896","11328","61184","182976","327168","349696","206848","52224"],["2448","35328","222432","797952","1783696","2543744","2259840","1143296","252160"],["6688","108480","780096","3264000","8756128","15616576","18515200","14070272","6218240","1217536"],["18272","329056","2660736","12720192","39813088","85239136","126413632","128221952","85121024","33394176","5878784"],["49920","988288","8875520","47725824","170723584","426556544","759538176","963796480","854048768","503298048","177512448","28385280"],["136384","2944000","29073088","173674752","698940736","1996244992","4148816960","6321627904","7008511744","5513193472","2920823808","935669760","137056256"],["372608","8709632","93802624","616299776","2755884672","8856610816","21043923840","37429571840","49832564224","49046508544","34685429760","16689672192","4897153024","661766144"],["1017984","25616000","298799488","2141374080","10533190784","37617593728","100585157760","204555976576","317925474560","375787243008","332511749120","213570119680","94124421120","25477160960","3195289600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","48","360","2736","20736","157248","1192320","9040896","68553216","519810048","3941498880","29886713856","226618269696","1718350184448"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["10","46","78","58","16"],["-8","-38","-68","-54","-16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","38","94","114","68","16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177741,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-8f2811a98b08dac6"]},{"id":177742,"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-8f2811a98b08dac6"]},{"id":177743,"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-8f2811a98b08dac6"]}],"region_exact":{},"incoming_edges":[{"id":"e-8f2811a98b08dac6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-0ac9d6c345261ad6","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"8ef6662eb5dd5a6a95645e711e61742c779f50bd4de66a6a845d64481e4be16d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-002f74523f94b1c9","role":"sequence","name":"basis_map_A007318 of Riordan array (1, 2+2x).","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-bda91c95ad1f8c7f","kind":"fingerprint_related","node_a_id":"n-0ac9d6c345261ad6","node_b_id":"n-9e6d672ceef3b9db","fingerprint":"f6-4600f723d5b687a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4600f723d5b687a0"},"other_id":"n-9e6d672ceef3b9db","other":{"id":"n-9e6d672ceef3b9db","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","first_index":1,"fingerprint":"f6-4600f723d5b687a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["6","22","20"]],"card_rows":[["1"],["2","4"],["6","22","20"],["16","88","160","96"]],"row_sums_preview":["1","6","48","360","2736","20736","157248","1192320"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-081205c1b1ff787d","e-6d01555a79e21a02"],"text":"A063967 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A063967 \u2192 basis A013609 \u2192 x \u21a6 x/(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:    1;\n  1:    2,      4;\n  2:    6,     22,     20;\n  3:   16,     88,    160,      96;\n  4:   44,    320,    868,    1040,     464;\n  5:  120,   1088,   3928,    7056,    6304,     2240;\n  6:  328,   3560,  16040,   38392,   51472,    36640,    10816;\n  7:  896,  11328,  61184,  182976,  327168,   349696,   206848,    52224;\n  8: 2448,  35328, 222432,  797952, 1783696,  2543744,  2259840,  1143296,  252160;\n  9: 6688, 108480, 780096, 3264000, 8756128, 15616576, 18515200, 14070272, 6218240, 1217536;\n","recurrence_sentence":"The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(8*n-10+(38*n-46)*x+(68*n-78)*x^2+(54*n-58)*x^3+(16*n-16)*x^4)*P(n-1,x) + (6+38*x+94*x^2+114*x^3+68*x^4+16*x^5)*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 - 10 + (38*n - 46)*x + (68*n - 78)*x^2 + (54*n - 58)*x^3 + (16*n - 16)*x^4)*P[n - 1, x] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*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] = (-(8*n - 10 + (38*n - 46)*x + (68*n - 78)*x^2 + (54*n - 58)*x^3 + (16*n - 16)*x^4)*v[i - 1] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); 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-f5a001953ee28e8f","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-23c3ede392db8800","e-8f2811a98b08dac6"],"text":"A099040 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A099040 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}