{"id":"n-f268708db386b729","revision":1,"first_index":1,"fingerprint":"f6-74459ae06326a5fd","name":"reversal 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"],["2","1"],["4","5","1"],["8","16","9","1"],["16","44","41","14","1"],["32","112","146","85","20","1"],["64","272","456","377","155","27","1"],["128","640","1312","1408","833","259","35","1"],["256","1472","3568","4712","3649","1652","406","44","1"],["512","3328","9312","14608","14002","8361","3024","606","54","1"],["1024","7424","23552","42800","48940","36365","17469","5202","870","65","1"],["2048","16384","58112","120064","159560","142000","85305","33957","8514","1210","77","1"],["4096","35840","140544","325376","492896","511380","369305","184690","62271","13376","1639","90","1"],["8192","77824","334336","857344","1458368","1728928","1459810","880685","374660","108823","20306","2171","104","1"],["16384","167936","784384","2206976","4165504","5556320","5373200","3800305","1956695","720005","182611","29939","2821","119","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","10","34","116","396","1352","4616","15760","53808","183712","627232","2141504","7311552","24963200"],"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":[["-2","-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":164199,"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-3cf80f21f053e148"]},{"id":164200,"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-3cf80f21f053e148"]},{"id":164201,"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-3cf80f21f053e148"]}],"region_exact":{},"incoming_edges":[{"id":"e-3cf80f21f053e148","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-f268708db386b729","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":"50c457e1b8c8275fea36c2118836280045f775433c8b9696eeac9faf1cdb1884","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-2821c21537bbbcdd","role":"sequence","name":"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-382b664e38a64523","kind":"fingerprint_related","node_a_id":"n-2821c21537bbbcdd","node_b_id":"n-f268708db386b729","fingerprint":"f6-74459ae06326a5fd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74459ae06326a5fd"},"other_id":"n-2821c21537bbbcdd","other":{"id":"n-2821c21537bbbcdd","name":"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":"f6-74459ae06326a5fd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","4"]],"card_rows":[["1"],["1","2"],["1","5","4"],["1","9","16","8"]],"row_sums_preview":["1","3","10","34","116","396","1352","4616"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":["A056242"],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A056242"],"edges":["e-3cb5f3d64c4bb317"],"text":"basis-map coefficients of A026729 \u2192 basis A056242"},"headline":"A056242: Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","chain":"basis-map coefficients of A026729 \u2192 basis A056242"}},{"id":"l-0ba8f654b98809ed","kind":"oeis_match","node_a_id":"n-f268708db386b729","node_b_id":null,"fingerprint":null,"oeis_id":"A048580","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A048580","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-3809635142d42de4","kind":"oeis_match","node_a_id":"n-f268708db386b729","node_b_id":null,"fingerprint":null,"oeis_id":"A007052","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A007052","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-63a76b89528b8eb9","kind":"oeis_match","node_a_id":"n-f268708db386b729","node_b_id":null,"fingerprint":null,"oeis_id":"A117317","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":58},"in_catalog":true,"oeis_id":"A117317","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c8b3582438c23ee9","kind":"oeis_match","node_a_id":"n-f268708db386b729","node_b_id":null,"fingerprint":null,"oeis_id":"A056242","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":59},"in_catalog":true,"oeis_id":"A056242","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A056242","relation":"match_reversed_triangle"},{"oeis_id":"A007052","relation":"match_row_sums"},{"oeis_id":"A048580","relation":"match_row_sums"},{"oeis_id":"A117317","relation":"match_triangle"}],"bfiles":[{"node_id":"n-f268708db386b729","oeis_id":"A007052","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":29,"oeis_terms_available":29,"term_count":200,"raw_size":11585},{"node_id":"n-f268708db386b729","oeis_id":"A048580","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":11691},{"node_id":"n-f268708db386b729","oeis_id":"A056242","target":"reversed_triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":59,"oeis_terms_agreeing":59,"oeis_terms_available":59,"term_count":990,"raw_size":16346},{"node_id":"n-f268708db386b729","oeis_id":"A117317","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":58,"oeis_terms_agreeing":58,"oeis_terms_available":58,"term_count":990,"raw_size":16344}],"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:   2,    1;\n   3:   4,    5,    1;\n   4:   8,   16,    9,     1;\n   5:  16,   44,   41,    14,     1;\n   6:  32,  112,  146,    85,    20,    1;\n   7:  64,  272,  456,   377,   155,   27,    1;\n   8: 128,  640, 1312,  1408,   833,  259,   35,   1;\n   9: 256, 1472, 3568,  4712,  3649, 1652,  406,  44,  1;\n  10: 512, 3328, 9312, 14608, 14002, 8361, 3024, 606, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(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 - 2 + (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 - 2 + (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}},{"oeis_id":"A056242","target":"reversed_triangle","first_row":1,"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: 1,  2;\n   3: 1,  5,   4;\n   4: 1,  9,  16,    8;\n   5: 1, 14,  41,   44,   16;\n   6: 1, 20,  85,  146,  112,    32;\n   7: 1, 27, 155,  377,  456,   272,    64;\n   8: 1, 35, 259,  833, 1408,  1312,   640,  128;\n   9: 1, 44, 406, 1652, 3649,  4712,  3568, 1472,  256;\n  10: 1, 54, 606, 3024, 8361, 14002, 14608, 9312, 3328, 512;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (2*n-2+(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 - 2 + (3*n - 5)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];\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] = ((2*n - 2 + (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 = 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":1,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A117317","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2,    1;\n  2:   4,    5,    1;\n  3:   8,   16,    9,     1;\n  4:  16,   44,   41,    14,     1;\n  5:  32,  112,  146,    85,    20,    1;\n  6:  64,  272,  456,   377,   155,   27,    1;\n  7: 128,  640, 1312,  1408,   833,  259,   35,   1;\n  8: 256, 1472, 3568,  4712,  3649, 1652,  406,  44,  1;\n  9: 512, 3328, 9312, 14608, 14002, 8361, 3024, 606, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (2*n+(3*n-2)*x)*P(n-1,x) - (2*x+2*x^2)*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[Together[((2*n + (3*n - 2)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = ((2*n + (3*n - 2)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A056242","reverse"],"edges":["e-3cb5f3d64c4bb317","e-3cf80f21f053e148"],"text":"basis-map coefficients of A026729 \u2192 basis A056242 \u2192 reverse"},"headline":"basis-map coefficients of A026729 \u2192 basis A056242 \u2192 reverse","chain":null}
