{"id":"n-4438653f9e73a43c","revision":1,"first_index":0,"fingerprint":"f6-629d13292f83019a","name":"reversal of Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-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","2"],["1","5","5"],["1","10","23","15"],["1","19","76","109","52"],["1","36","224","531","544","203"],["1","69","631","2204","3641","2876","877"],["1","134","1749","8471","20089","25208","16113","4140"],["1","263","4838","31331","100171","177631","178564","95495","21147"],["1","520","13454","113807","471827","1109473","1559788","1302451","597155","115975"],["1","1033","37753","410706","2151150","6445823","11887800","13780656","9812188","3929243","678570"],["1","2058","107051","1482515","9632314","35799996","83199346","125567333","123442428","76447966","27132324","4213597"],["1","4107","306736","5374225","42750566","193307542","550769220","1039336726","1322821787","1126307324","616159974","196122796","27644437"],["1","8204","887340","19611307","189188570","1025976095","3513380012","8063079454","12758127667","14003320332","10496931158","5136151320","1480531285","190899322"],["1","16397","2588171","72135400","838125575","5392177790","21864986859","59810206884","114418727281","155445364931","149698846727","100092914934","44253777751","11647194573","1382958545"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","11","49","257","1539","10299","75905","609441","5284451","49134923","487026929","5120905441","56878092067","664920021819"],"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","3","1"],["0","-1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","2","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165819,"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-0bec24e5cc539db1"]},{"id":165820,"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-0bec24e5cc539db1"]},{"id":165821,"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-0bec24e5cc539db1"]}],"region_exact":{},"incoming_edges":[{"id":"e-0bec24e5cc539db1","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-4438653f9e73a43c","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":"f42f8f80f08666b846f2e47df691048b28464de5f5d6ac5ba68b89d7e8b23c03","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-8f767a3615460a6b","role":"sequence","name":"Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-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-5fa98aaa2f86b33c","kind":"fingerprint_related","node_a_id":"n-c96d9794dc287517","node_b_id":"n-4438653f9e73a43c","fingerprint":"f6-629d13292f83019a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-629d13292f83019a"},"other_id":"n-c96d9794dc287517","other":{"id":"n-c96d9794dc287517","name":"mobius_x_over_x_plus_one of Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-629d13292f83019a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","10","23","15"]],"row_sums_preview":["1","3","11","49","257","1539","10299","75905"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["x \u21a6 x/(x+1)"],"edges":["e-6e9c730f2a536c2a"],"text":"A008277 \u2192 x \u21a6 x/(x+1)"},"headline":"A008277 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-78f746acbdc479ac","kind":"fingerprint_related","node_a_id":"n-796737d79315d8c3","node_b_id":"n-4438653f9e73a43c","fingerprint":"f6-629d13292f83019a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-629d13292f83019a"},"other_id":"n-796737d79315d8c3","other":{"id":"n-796737d79315d8c3","name":"shift_x_plus_one of Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-629d13292f83019a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["15","23","10","1"]],"row_sums_preview":["1","3","11","49","257","1539","10299","75905"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f277d949034445ba","steps":["x \u21a6 x+1"],"edges":["e-310068e829fb9797"],"text":"A008278 \u2192 x \u21a6 x+1"},"headline":"A008278 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-7b9253f358e9bde0","kind":"fingerprint_related","node_a_id":"n-95b4aa73235b5230","node_b_id":"n-4438653f9e73a43c","fingerprint":"f6-629d13292f83019a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-629d13292f83019a"},"other_id":"n-95b4aa73235b5230","other":{"id":"n-95b4aa73235b5230","name":"mobius_x_over_x_plus_one of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","first_index":0,"fingerprint":"f6-629d13292f83019a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","2"]],"card_rows":[["1"],["0","1"],["0","1","2"],["0","1","5","5"]],"row_sums_preview":["1","1","3","11","49","257","1539","10299"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ee613bf6b082e7d","steps":["reverse (shift +1)","x \u21a6 x/(x+1)"],"edges":["e-ad13a5d17ca116d8","e-ef1c37b6ab776d5a"],"text":"A106800 \u2192 reverse (shift +1) \u2192 x \u21a6 x/(x+1)"},"headline":"A106800 \u2192 reverse (shift +1) \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-d3c26053ad7ea178","kind":"fingerprint_related","node_a_id":"n-c6b873425efe3e00","node_b_id":"n-4438653f9e73a43c","fingerprint":"f6-629d13292f83019a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-629d13292f83019a"},"other_id":"n-c6b873425efe3e00","other":{"id":"n-c6b873425efe3e00","name":"shift_x_plus_one of Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.","first_index":1,"fingerprint":"f6-629d13292f83019a","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["5","5","1"]],"row_sums_preview":["1","1","3","11","49","257","1539","10299"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ee613bf6b082e7d","steps":["x \u21a6 x+1"],"edges":["e-af1b637d986229c4"],"text":"A106800 \u2192 x \u21a6 x+1"},"headline":"A106800 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-f4c6ba43c972d255","kind":"fingerprint_related","node_a_id":"n-8f767a3615460a6b","node_b_id":"n-4438653f9e73a43c","fingerprint":"f6-629d13292f83019a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-629d13292f83019a"},"other_id":"n-8f767a3615460a6b","other":{"id":"n-8f767a3615460a6b","name":"Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.","first_index":0,"fingerprint":"f6-629d13292f83019a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["15","23","10","1"]],"row_sums_preview":["1","3","11","49","257","1539","10299","75905"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":["A059340"],"derivation":{"root":"n-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)"],"edges":["e-236d0fb15dc6eca0"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1)"},"headline":"A059340: Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.","chain":"A008278 \u2192 x \u21a6 x+1 (shift +1)"}},{"id":"l-9fd31d1ab593daaf","kind":"oeis_match","node_a_id":"n-4438653f9e73a43c","node_b_id":null,"fingerprint":null,"oeis_id":"A059340","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A059340","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a0e6b3da12880c38","kind":"oeis_match","node_a_id":"n-4438653f9e73a43c","node_b_id":null,"fingerprint":null,"oeis_id":"A004211","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A004211","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A059340","relation":"match_reversed_triangle"},{"oeis_id":"A004211","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-4438653f9e73a43c","oeis_id":"A059340","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":29697}],"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,   2;\n  2: 1,   5,     5;\n  3: 1,  10,    23,     15;\n  4: 1,  19,    76,    109,     52;\n  5: 1,  36,   224,    531,    544,     203;\n  6: 1,  69,   631,   2204,   3641,    2876,     877;\n  7: 1, 134,  1749,   8471,  20089,   25208,   16113,    4140;\n  8: 1, 263,  4838,  31331, 100171,  177631,  178564,   95495,  21147;\n  9: 1, 520, 13454, 113807, 471827, 1109473, 1559788, 1302451, 597155, 115975;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1-(n-3)*x-(n-1)*x^2)*P(n-1,x) + (x+2*x^2+x^3)*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[(1 - (n - 3)*x - (n - 1)*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], x]];\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 - 3)*x - (n - 1)*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*deriv(v[i - 1], 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}},{"oeis_id":"A059340","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      2,      1;\n   3:      5,      5,       1;\n   4:     15,     23,      10,       1;\n   5:     52,    109,      76,      19,       1;\n   6:    203,    544,     531,     224,      36,      1;\n   7:    877,   2876,    3641,    2204,     631,     69,      1;\n   8:   4140,  16113,   25208,   20089,    8471,   1749,    134,     1;\n   9:  21147,  95495,  178564,  177631,  100171,  31331,   4838,   263,   1;\n  10: 115975, 597155, 1302451, 1559788, 1109473, 471827, 113807, 13454, 520, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1-(n-4)*x-(n-2)*x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx 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 - (n - 4)*x - (n - 2)*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 - (n - 4)*x - (n - 2)*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","reverse"],"edges":["e-236d0fb15dc6eca0","e-0bec24e5cc539db1"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 reverse"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 reverse","chain":null}