{"id":"n-900aa85ccaa9b50d","revision":0,"first_index":0,"fingerprint":"f6-aa231a1ef2aefcc4","name":"reversal of theta_plus_one_squared of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4"],["9","4"],["16","27","4"],["25","96","63","4"],["36","250","400","135","4"],["49","540","1625","1440","279","4"],["64","1029","5040","8750","4816","567","4"],["81","1792","13034","37800","42525","15456","1143","4"],["100","2916","29568","129654","250236","194250","48400","2295","4"],["121","4500","60750","376320","1118523","1530900","852625","149280","4599","4"],["144","6655","115500","962280","4095168","8794863","8882280","3643750","456016","9207","4"],["169","9504","206305","2227500","12881187","40153344","64858948","49658400","15287525","1384416","18423","4"],["196","13182","350064","4758325","35950200","153868572","365787136","456744288","270306036","63313250","4186000","36855","4"],["225","17836","569023","9513504","91082992","513513000","1693897920","3157073920","3108382277","1442701260","259793625","12623520","73719","4"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","4","13","47","188","825","3937","20270","111835","657423","4097622","26965867","186685725","1355314108","10289242825"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","24","-6"],["4","-20","5"],["0","4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4","24","-6"],["0","0","-8","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","4","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["8","-3"],["-4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176997,"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-0c4b27ae8a3dfbb4"]},{"id":176998,"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-0c4b27ae8a3dfbb4"]},{"id":176999,"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-0c4b27ae8a3dfbb4"]}],"region_exact":{},"incoming_edges":[{"id":"e-0c4b27ae8a3dfbb4","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-900aa85ccaa9b50d","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":"63b4a7f93dbc2b1078811b14312f416805b59d9d8572cd6401150e69db1c6d5b","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-6def5b7e0013bbda","role":"sequence","name":"theta_plus_one_squared of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","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-dc58f9748d3d94b2","kind":"fingerprint_related","node_a_id":"n-6def5b7e0013bbda","node_b_id":"n-900aa85ccaa9b50d","fingerprint":"f6-aa231a1ef2aefcc4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-aa231a1ef2aefcc4"},"other_id":"n-6def5b7e0013bbda","other":{"id":"n-6def5b7e0013bbda","name":"theta_plus_one_squared of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","first_index":0,"fingerprint":"f6-aa231a1ef2aefcc4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","4"],["0","4","9"]],"card_rows":[["1"],["0","4"],["0","4","9"],["0","4","27","16"]],"row_sums_preview":["1","4","13","47","188","825","3937","20270"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e65f982fa1b436c5","steps":["(\u03b8+1)\u00b2"],"edges":["e-61c911ef7d81056e"],"text":"A048993 \u2192 (\u03b8+1)\u00b2"},"headline":"A048993 \u2192 (\u03b8+1)\u00b2","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;\n  2:   9,    4;\n  3:  16,   27,     4;\n  4:  25,   96,    63,      4;\n  5:  36,  250,   400,    135,      4;\n  6:  49,  540,  1625,   1440,    279,      4;\n  7:  64, 1029,  5040,   8750,   4816,    567,     4;\n  8:  81, 1792, 13034,  37800,  42525,  15456,  1143,    4;\n  9: 100, 2916, 29568, 129654, 250236, 194250, 48400, 2295, 4;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+1+(4*n^2-20*n+24)*x-(n^2-5*n+6)*x^2)*P(n-2,x) - (4*x+(8*n-24)*x^2-(2*n-6)*x^3)*d/dx P(n-2,x) + (4*x^3-x^4)*d^2/dx^2 P(n-2,x) - (4*n-8-(2*n-3)*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[(4*n + 1 + (4*n^2 - 20*n + 24)*x - (n^2 - 5*n + 6)*x^2)*P[n - 2, x] - (4*x + (8*n - 24)*x^2 - (2*n - 6)*x^3)*D[P[n - 2, x], x] + (4*x^3 - x^4)*D[P[n - 2, x], {x, 2}] - (4*n - 8 - (2*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*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; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (4*n + 1 + (4*n^2 - 20*n + 24)*x - (n^2 - 5*n + 6)*x^2)*v[i - 2] - (4*x + (8*n - 24)*x^2 - (2*n - 6)*x^3)*deriv(v[i - 2], x) + (4*x^3 - x^4)*deriv(deriv(v[i - 2], x), x) - (4*n - 8 - (2*n - 3)*x)*v[i - 1] + (4*x - 2*x^2)*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}}],"derivation":{"root":"n-e65f982fa1b436c5","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-61c911ef7d81056e","e-0c4b27ae8a3dfbb4"],"text":"A048993 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"A048993 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}
