{"id":"n-e9df25e24f0e0fe4","revision":0,"first_index":0,"fingerprint":"f6-c9cad0c5368eab69","name":"reversal of theta_plus_one of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","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","1"],["3","8","1"],["4","33","22","1"],["5","96","210","48","1"],["6","225","1256","942","90","1"],["7","456","5395","10912","3237","152","1"],["8","833","18270","84975","67980","9135","238","1"],["9","1408","51940","482784","933790","321856","22260","352","1"],["10","2241","129312","2149868","8945172","7454310","1228496","48492","498","1"],["11","3400","290277","7928704","63278222","118739664","45198730","3964352","96759","680","1"],["12","4961","599830","25212429","350515792","1349771570","1156947060","219072370","11205524","179949","902","1"],["13","7008","1158454","71187600","1596955239","11439083648","20985451396","8579312736","887197355","28475040","315942","1168","1"],["14","9633","2115048","182395642","6215736670","75981445839","276067445152","241559014508","50654297226","3107868335","66325688","528762","1482","1"],["15","12936","3682679","431097888","21271463499","413579402600","2728709698563","4917108780672","2122329765549","248147641560","9668847045","143699296","849849","1848","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","60","360","2520","20160","181440","1814400","19958400","239500800","3113510400","43589145600","653837184000","10461394944000"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"],["3","8","1"],["4","33","22","1"]],"recurrence":{"denominator":{"coeffs":[["3"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-12"],["0","0","40"],["0","0","-63"],["0","0","59"],["0","0","-32"],["0","0","9"],["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-4"],["0","20","10"],["0","-20","-12"],["0","10","8"],["0","-2","-2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-3","-2","-1"],["3","-4","1"],["-1","10","-1"],["0","-8","0"],["0","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","4"],["-6","-4"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181527,"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-92d62bcad206d4fd"]},{"id":181528,"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-92d62bcad206d4fd"]},{"id":181529,"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-92d62bcad206d4fd"]}],"region_exact":{},"incoming_edges":[{"id":"e-92d62bcad206d4fd","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-e9df25e24f0e0fe4","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":"614a0bf01c1c9e59ad8a9f896d79019c289f98a78c162a9ade1538513efdf3f2","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-2a3a7c185d653866","role":"sequence","name":"theta_plus_one of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","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-68dece81786983cd","kind":"fingerprint_related","node_a_id":"n-2a3a7c185d653866","node_b_id":"n-e9df25e24f0e0fe4","fingerprint":"f6-c9cad0c5368eab69","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c9cad0c5368eab69"},"other_id":"n-2a3a7c185d653866","other":{"id":"n-2a3a7c185d653866","name":"theta_plus_one of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","first_index":0,"fingerprint":"f6-c9cad0c5368eab69","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","8","3"]],"card_rows":[["1"],["1","2"],["1","8","3"],["1","22","33","4"]],"row_sums_preview":["1","3","12","60","360","2520","20160","181440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08ac2c6d9914754c","steps":["\u03b8+1"],"edges":["e-1ef33fb3ca062b95"],"text":"A154986 \u2192 \u03b8+1"},"headline":"A154986 \u2192 \u03b8+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,    1;\n  2:  3,    8,      1;\n  3:  4,   33,     22,       1;\n  4:  5,   96,    210,      48,       1;\n  5:  6,  225,   1256,     942,      90,       1;\n  6:  7,  456,   5395,   10912,    3237,     152,       1;\n  7:  8,  833,  18270,   84975,   67980,    9135,     238,     1;\n  8:  9, 1408,  51940,  482784,  933790,  321856,   22260,   352,   1;\n  9: 10, 2241, 129312, 2149868, 8945172, 7454310, 1228496, 48492, 498, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+3)*P(n,x) = -(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^2*P(n-4,x) - ((2*n^4-10*n^3+20*n^2-20*n+8)*x+(2*n^4-8*n^3+12*n^2-10*n+4)*x^2)*P(n-3,x) - (n^2-3*n+3-(2*n^4-8*n^3+10*n^2-4*n-2)*x+(n^2-n+1)*x^2)*P(n-2,x) + (2*n^2-6*n+6+(2*n^2-4*n+4)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 3 + 8*x + x^2, P(3,x) = 4 + 33*x + 22*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 3 + 8*x + x^2;\nP[3, x_] := 4 + 33*x + 22*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2*P[n - 4, x] - ((2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x + (2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x^2)*P[n - 3, x] - (n^2 - 3*n + 3 - (2*n^4 - 8*n^3 + 10*n^2 - 4*n - 2)*x + (n^2 - n + 1)*x^2)*P[n - 2, x] + (2*n^2 - 6*n + 6 + (2*n^2 - 4*n + 4)*x)*P[n - 1, x])/(n^2 - 3*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 3 + 8*x + x^2; v[4] = 4 + 33*x + 22*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2*v[i - 4] - ((2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x + (2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x^2)*v[i - 3] - (n^2 - 3*n + 3 - (2*n^4 - 8*n^3 + 10*n^2 - 4*n - 2)*x + (n^2 - n + 1)*x^2)*v[i - 2] + (2*n^2 - 6*n + 6 + (2*n^2 - 4*n + 4)*x)*v[i - 1])/(n^2 - 3*n + 3)); 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-08ac2c6d9914754c","steps":["\u03b8+1","reverse"],"edges":["e-1ef33fb3ca062b95","e-92d62bcad206d4fd"],"text":"A154986 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A154986 \u2192 \u03b8+1 \u2192 reverse","chain":null}
