{"id":"n-f623090076ebd096","revision":0,"first_index":1,"fingerprint":"f6-766b2adfc13b2af9","name":"reversal of theta_plus_one of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","1"],["12","10","1"],["32","48","18","1"],["80","176","123","28","1"],["192","560","584","255","40","1"],["448","1632","2280","1508","465","54","1"],["1024","4480","7872","7040","3332","777","70","1"],["2304","11776","24976","28272","18245","6608","1218","88","1"],["5120","29952","74496","102256","84012","41805","12096","1818","108","1"],["11264","74240","211968","342400","342580","218190","87345","20808","2610","130","1"],["24576","180224","581120","1080576","1276480","994000","511830","169785","34056","3630","154","1"],["53248","430080","1545984","3253760","4436064","4091040","2585135","1108140","311355","53504","4917","180","1"],["114688","1011712","4012032","9430784","14583680","15560352","11678480","6164795","2247960","544115","81224","6513","208","1"],["245760","2351104","10196992","26483712","45820544","55563200","48358800","30402440","13696865","4320030","913055","119756","8463","238","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","23","99","408","1632","6388","24596","93488","351664","1311536","4856432","17873408","65436544","238480960"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-5"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175882,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-47088e65c22f45c1"]},{"id":175883,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-47088e65c22f45c1"]},{"id":175884,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-47088e65c22f45c1"]}],"region_exact":{},"incoming_edges":[{"id":"e-47088e65c22f45c1","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-f623090076ebd096","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":"855a9f9d283817846e3838fd12555b253bb053cf72680c0b75ab457c46bf098b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2d155baa50423f8c","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-729fde3a5ca99d2a","kind":"fingerprint_related","node_a_id":"n-2d155baa50423f8c","node_b_id":"n-f623090076ebd096","fingerprint":"f6-766b2adfc13b2af9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-766b2adfc13b2af9"},"other_id":"n-2d155baa50423f8c","other":{"id":"n-2d155baa50423f8c","name":"theta_plus_one of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","first_index":1,"fingerprint":"f6-766b2adfc13b2af9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","10","12"]],"card_rows":[["1"],["1","4"],["1","10","12"],["1","18","48","32"]],"row_sums_preview":["1","5","23","99","408","1632","6388","24596"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2821c21537bbbcdd","steps":["\u03b8+1"],"edges":["e-0732c8d06b4db3af"],"text":"A056242 \u2192 \u03b8+1"},"headline":"A056242 \u2192 \u03b8+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    4,     1;\n   3:   12,    10,     1;\n   4:   32,    48,    18,      1;\n   5:   80,   176,   123,     28,     1;\n   6:  192,   560,   584,    255,    40,     1;\n   7:  448,  1632,  2280,   1508,   465,    54,     1;\n   8: 1024,  4480,  7872,   7040,  3332,   777,    70,    1;\n   9: 2304, 11776, 24976,  28272, 18245,  6608,  1218,   88,   1;\n  10: 5120, 29952, 74496, 102256, 84012, 41805, 12096, 1818, 108, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n+(3*n-5)*x)*P(n-1,x) - (2*x+2*x^2)*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[Together[((2*n + (3*n - 5)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n + (3*n - 5)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2821c21537bbbcdd","steps":["\u03b8+1","reverse"],"edges":["e-0732c8d06b4db3af","e-47088e65c22f45c1"],"text":"A056242 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A056242 \u2192 \u03b8+1 \u2192 reverse","chain":null}
