{"id":"n-5e246db2a215203d","revision":0,"first_index":0,"fingerprint":"f6-371e17a42c112769","name":"reversal of basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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"],["3","4"],["6","18","13"],["12","57","87","43"],["24","156","369","378","142"],["48","396","1275","2007","1548","469"],["96","960","3918","8364","9864","6102","1549"],["192","2256","11160","30153","48102","45351","23421","5116"],["384","5184","30144","98664","198945","253236","198861","88146","16897"],["768","11712","78288","301176","735195","1181562","1250883","841647","326769","55807"],["1536","26112","197280","872544","2502726","4866318","6498633","5888040","3465486","1196964","184318"],["3072","57600","485376","2426928","8002548","18277185","29512977","33704424","26687622","13958802","4342104","608761"],["6144","125952","1171200","6533376","24355704","63937284","121226553","167312730","166875189","117333996","55223064","15625116","2010601"],["12288","273408","2780928","17123328","71223840","211371156","460544691","746095281","898905969","795685605","503094540","215220330","55847415","6640564"],["24576","589824","6514176","43886592","201522240","667302912","1643495838","3058988472","4324637064","4622277036","3677455080","2112281340","828204000","198459090","21932293"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","37","199","1069","5743","30853","165751","890461","4783807","25699957","138067399","741736909","3984819343","21407570533"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["-4","2","34","26"],["4","-9","-50","-39"],["0","7","16","13"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","44","92","52"],["0","-4","-24","-45","-26"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","4","20","29","13"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178872,"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-ce94004ae3dbb148"]},{"id":178873,"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-ce94004ae3dbb148"]},{"id":178874,"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-ce94004ae3dbb148"]}],"region_exact":{},"incoming_edges":[{"id":"e-ce94004ae3dbb148","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-5e246db2a215203d","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":"fc26bb562afe6dab5fe8d1e7650d642f20412ad291cbef88d70ccfbfe26e187c","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-359eaa7b22014e19","role":"sequence","name":"basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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-5acd7fc47fd3b2d3","kind":"candidate_equal","node_a_id":"n-a0e65e4dc3f9251c","node_b_id":"n-5e246db2a215203d","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-a0e65e4dc3f9251c","other":{"id":"n-a0e65e4dc3f9251c","name":"mobius_x_over_x_plus_one of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","first_index":0,"fingerprint":"f6-371e17a42c112769","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","4"],["6","18","13"]],"card_rows":[["1"],["3","4"],["6","18","13"],["12","57","87","43"]],"row_sums_preview":["1","7","37","199","1069","5743","30853","165751"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6b7687ba0ff59dd","steps":["x \u21a6 x/(x+1)"],"edges":["e-36a264a0f06b658e"],"text":"A116412 \u2192 x \u21a6 x/(x+1)"},"headline":"A116412 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-2e4daba00a5d0a13","kind":"fingerprint_related","node_a_id":"n-359eaa7b22014e19","node_b_id":"n-5e246db2a215203d","fingerprint":"f6-371e17a42c112769","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-371e17a42c112769"},"other_id":"n-359eaa7b22014e19","other":{"id":"n-359eaa7b22014e19","name":"basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","first_index":0,"fingerprint":"f6-371e17a42c112769","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","3"],["13","18","6"]],"card_rows":[["1"],["4","3"],["13","18","6"],["43","87","57","12"]],"row_sums_preview":["1","7","37","199","1069","5743","30853","165751"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e7331b0d50084c15","steps":["basis A007318"],"edges":["e-212843e2c5c5a487"],"text":"A249251 \u2192 basis A007318"},"headline":"A249251 \u2192 basis A007318","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:   3,     4;\n  2:   6,    18,    13;\n  3:  12,    57,    87,     43;\n  4:  24,   156,   369,    378,    142;\n  5:  48,   396,  1275,   2007,   1548,     469;\n  6:  96,   960,  3918,   8364,   9864,    6102,    1549;\n  7: 192,  2256, 11160,  30153,  48102,   45351,   23421,   5116;\n  8: 384,  5184, 30144,  98664, 198945,  253236,  198861,  88146,  16897;\n  9: 768, 11712, 78288, 301176, 735195, 1181562, 1250883, 841647, 326769, 55807;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (4*n-4+(7*n^2-9*n+2)*x+(16*n^2-50*n+34)*x^2+(13*n^2-39*n+26)*x^3)*P(n-1,x) - ((4*n-4)*x+(24*n-44)*x^2+(45*n-92)*x^3+(26*n-52)*x^4)*d/dx P(n-1,x) + (4*x^2+20*x^3+29*x^4+13*x^5)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (7*n^2 - 9*n + 2)*x + (16*n^2 - 50*n + 34)*x^2 + (13*n^2 - 39*n + 26)*x^3)*P[n - 1, x] - ((4*n - 4)*x + (24*n - 44)*x^2 + (45*n - 92)*x^3 + (26*n - 52)*x^4)*D[P[n - 1, x], x] + (4*x^2 + 20*x^3 + 29*x^4 + 13*x^5)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (7*n^2 - 9*n + 2)*x + (16*n^2 - 50*n + 34)*x^2 + (13*n^2 - 39*n + 26)*x^3)*v[i - 1] - ((4*n - 4)*x + (24*n - 44)*x^2 + (45*n - 92)*x^3 + (26*n - 52)*x^4)*deriv(v[i - 1], x) + (4*x^2 + 20*x^3 + 29*x^4 + 13*x^5)*deriv(deriv(v[i - 1], x), x))/(2*n - 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-e7331b0d50084c15","steps":["basis A007318","reverse"],"edges":["e-212843e2c5c5a487","e-ce94004ae3dbb148"],"text":"A249251 \u2192 basis A007318 \u2192 reverse"},"headline":"A249251 \u2192 basis A007318 \u2192 reverse","chain":null}