{"id":"n-6c47aa5f31984af6","revision":0,"first_index":0,"fingerprint":"f6-679f9f4065461fd6","name":"reversal of basis_map_A013611 of Pascal-(1,5,1) array.","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","2"],["16","36","9"],["64","256","168","28"],["256","1472","1984","808","101"],["1024","7680","16320","11840","3420","342"],["4096","37888","111360","127040","63440","14268","1189"],["16384","180224","679936","1095680","826240","311552","57232","4088"],["65536","835584","3858432","8159232","8593920","4802688","1454592","225936","14121"],["262144","3801088","20791296","54853632","76007424","58401792","25785984","6515328","876276","48682"],["1048576","17039360","107806720","342343680","596213760","598296576","358970880","130738560","28327440","3359380","167969"],["4194304","75497472","542638080","2019819520","4270571520","5380030464","4180549632","2048624640","634023360","120194560","12745656","579348"],["16777216","331350016","2667577344","11405885440","28507832320","43697258496","42429259776","26726498304","11042968320","2969066560","500196544","47964984","1998541"],["67108864","1442840576","12863930368","62188945408","179952680960","327315226624","386065022976","302275657728","159500110848","56885793280","13512927296","2048138944","179239372","6893822"],["268435456","6241124352","61052289024","329491939328","1085644210176","2296352342016","3215374876672","3048304164864","1989228023808","901035252736","282419468544","60067674048","8273517776","665849772","23780349"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","61","516","4621","40626","359281","3171336","28010041","247343646","2184312901","19289468556","170344633861","1504304517066","13284442948921"],"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"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0","18","6"],["8","-10","-30","-9"],["0","14","12","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","74","66","12"],["0","-8","-52","-36","-6"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","8","50","24","3"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184762,"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-e211f65c14b4f763"]},{"id":184763,"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-e211f65c14b4f763"]},{"id":184764,"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-e211f65c14b4f763"]}],"region_exact":{},"incoming_edges":[{"id":"e-e211f65c14b4f763","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-6c47aa5f31984af6","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":"28e7416b34c331ddbee7987ac9449b726f58232157cd2e92cde4ab14ed019bc0","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-590ad09c6151a6a8","role":"sequence","name":"basis_map_A013611 of Pascal-(1,5,1) array.","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-92529cc2a3f6f69e","kind":"fingerprint_related","node_a_id":"n-590ad09c6151a6a8","node_b_id":"n-6c47aa5f31984af6","fingerprint":"f6-679f9f4065461fd6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-679f9f4065461fd6"},"other_id":"n-590ad09c6151a6a8","other":{"id":"n-590ad09c6151a6a8","name":"basis_map_A013611 of Pascal-(1,5,1) array.","first_index":0,"fingerprint":"f6-679f9f4065461fd6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["9","36","16"]],"card_rows":[["1"],["2","4"],["9","36","16"],["28","168","256","64"]],"row_sums_preview":["1","6","61","516","4621","40626","359281","3171336"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6ad01dc6e705cdb6","steps":["basis A013611"],"edges":["e-09b3cf169dcfe004"],"text":"A081580 \u2192 basis A013611"},"headline":"A081580 \u2192 basis A013611","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,       2;\n  2:     16,      36,        9;\n  3:     64,     256,      168,       28;\n  4:    256,    1472,     1984,      808,      101;\n  5:   1024,    7680,    16320,    11840,     3420,      342;\n  6:   4096,   37888,   111360,   127040,    63440,    14268,     1189;\n  7:  16384,  180224,   679936,  1095680,   826240,   311552,    57232,    4088;\n  8:  65536,  835584,  3858432,  8159232,  8593920,  4802688,  1454592,  225936,  14121;\n  9: 262144, 3801088, 20791296, 54853632, 76007424, 58401792, 25785984, 6515328, 876276, 48682;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = (8*n+(14*n^2-10*n)*x+(12*n^2-30*n+18)*x^2+(3*n^2-9*n+6)*x^3)*P(n-1,x) - (8*n*x+(52*n-74)*x^2+(36*n-66)*x^3+(6*n-12)*x^4)*d/dx P(n-1,x) + (8*x^2+50*x^3+24*x^4+3*x^5)*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[Together[((8*n + (14*n^2 - 10*n)*x + (12*n^2 - 30*n + 18)*x^2 + (3*n^2 - 9*n + 6)*x^3)*P[n - 1, x] - (8*n*x + (52*n - 74)*x^2 + (36*n - 66)*x^3 + (6*n - 12)*x^4)*D[P[n - 1, x], x] + (8*x^2 + 50*x^3 + 24*x^4 + 3*x^5)*D[P[n - 1, x], {x, 2}])/(2*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 + (14*n^2 - 10*n)*x + (12*n^2 - 30*n + 18)*x^2 + (3*n^2 - 9*n + 6)*x^3)*v[i - 1] - (8*n*x + (52*n - 74)*x^2 + (36*n - 66)*x^3 + (6*n - 12)*x^4)*deriv(v[i - 1], x) + (8*x^2 + 50*x^3 + 24*x^4 + 3*x^5)*deriv(deriv(v[i - 1], x), x))/(2*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-6ad01dc6e705cdb6","steps":["basis A013611","reverse"],"edges":["e-09b3cf169dcfe004","e-e211f65c14b4f763"],"text":"A081580 \u2192 basis A013611 \u2192 reverse"},"headline":"A081580 \u2192 basis A013611 \u2192 reverse","chain":null}
