{"id":"n-2bcf501d4853538e","revision":0,"first_index":0,"fingerprint":"f6-ca6d936c14642db2","name":"mobius_x_over_x_plus_one of shift_x_plus_one of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+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"],["4","6"],["15","45","33"],["56","252","372","180"],["210","1260","2800","2730","985"],["792","5940","17640","25920","18840","5418"],["3003","27027","100485","197505","216405","125307","29953"],["11440","120120","536536","1321320","1937320","1690920","813400","166344"],["43758","525096","2738736","8108100","14900886","17405388","12618144","5190372","927441"],["167960","2267460","13525200","46781280","103390560","151400340","146884920","91035360","32704380","5188590"],["646646","9699690","65126490","257734620","665705040","1172539368","1426184760","1182791610","640074435","204086025","29113953"],["2496144","41186376","307405560","1369894680","4049567280","8337555072","12199084128","12683895840","9183802320","4409933220","1263896172","163786428"],["9657700","173838600","1427794368","7075231020","23557775670","55521025560","94968279120","118784066544","107820149580","69262787880","29889082080","7779300750","923511849"],["37442160","730122120","6544057520","35694859200","132189961904","350969703084","687281234640","1005012040032","1097399647344","883703837016","510095874288","199860321024","47640373872","5217709266"],["145422675","3053876175","29661211125","176598287775","720032237925","2126632783275","4691977695405","7855370851185","10028281548285","9714215285775","7028458421985","3683073257955","1321375611465","290519829495","29532022785"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","93","860","7985","74550","699685","6597400","62457921","593346050","5653702637","54012503220","517192500721","4962377183470","47698928343285"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["2","4","2"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["2","4","-2","-8","-4"],["6","21","24","9","0"],["4","17","26","17","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-10","-18","-14","-4"],["0","-2","-10","-18","-14","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179743,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-6d171e78219b20a8"]},{"id":179744,"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-6d171e78219b20a8"]},{"id":179745,"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-6d171e78219b20a8"]}],"region_exact":{},"incoming_edges":[{"id":"e-6d171e78219b20a8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-2bcf501d4853538e","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"c3f04d1daa09a3a322c4185bb78836948754218e1d97adafe22ed153852e1d22","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-00a555c278779f26","role":"sequence","name":"shift_x_plus_one of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+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-1a1cd1b0ca15c71a","kind":"fingerprint_related","node_a_id":"n-ff3f26cc45b3d80a","node_b_id":"n-2bcf501d4853538e","fingerprint":"f6-ca6d936c14642db2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ca6d936c14642db2"},"other_id":"n-ff3f26cc45b3d80a","other":{"id":"n-ff3f26cc45b3d80a","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","first_index":0,"fingerprint":"f6-ca6d936c14642db2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["5","15","11"]],"card_rows":[["1"],["2","3"],["5","15","11"],["14","63","93","45"]],"row_sums_preview":["1","5","31","215","1597","12425","99955","824675"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-57b4b14604f4c0d7","steps":["x \u21a6 x/(x+1)"],"edges":["e-38a5f21f7e3f9e41"],"text":"A126216 \u2192 x \u21a6 x/(x+1)"},"headline":"A126216 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-512d63154aece76c","kind":"fingerprint_related","node_a_id":"n-2bcf501d4853538e","node_b_id":"n-6a8ebfe1e41f2334","fingerprint":"f6-ca6d936c14642db2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ca6d936c14642db2"},"other_id":"n-6a8ebfe1e41f2334","other":{"id":"n-6a8ebfe1e41f2334","name":"derivative of basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-ca6d936c14642db2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["18","12"]],"card_rows":[["2"],["18","12"],["132","180","60"],["900","1860","1260","280"]],"row_sums_preview":["0","2","30","372","4300","47910","521850","5597480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4689fb1386d440ad","steps":["x \u21a6 x+1","D"],"edges":["e-437a2dd956a9df17","e-852e18e544fa01dc"],"text":"A063007 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A063007 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-704f5162ed88d7fa","kind":"fingerprint_related","node_a_id":"n-2bcf501d4853538e","node_b_id":"n-c0e24188ccfeaae9","fingerprint":"f6-ca6d936c14642db2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ca6d936c14642db2"},"other_id":"n-c0e24188ccfeaae9","other":{"id":"n-c0e24188ccfeaae9","name":"reversal of basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","first_index":0,"fingerprint":"f6-ca6d936c14642db2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["5","15","11"]],"card_rows":[["1"],["2","3"],["5","15","11"],["14","63","93","45"]],"row_sums_preview":["1","5","31","215","1597","12425","99955","824675"],"regions":["(-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-66e2c3448245b0b7","steps":["x \u21a6 x+1","reverse"],"edges":["e-190c283dd8b47413","e-e5739c6a1dde738f"],"text":"A033282 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A033282 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-9cca44a48d90549e","kind":"fingerprint_related","node_a_id":"n-07af2e6e3c9f0c77","node_b_id":"n-2bcf501d4853538e","fingerprint":"f6-ca6d936c14642db2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ca6d936c14642db2"},"other_id":"n-07af2e6e3c9f0c77","other":{"id":"n-07af2e6e3c9f0c77","name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","first_index":0,"fingerprint":"f6-ca6d936c14642db2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["11","15","5"]],"card_rows":[["1"],["3","2"],["11","15","5"],["45","93","63","14"]],"row_sums_preview":["1","5","31","215","1597","12425","99955","824675"],"regions":["(-inf, 0]","(-2, -1)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-66e2c3448245b0b7","steps":["x \u21a6 x+1"],"edges":["e-190c283dd8b47413"],"text":"A033282 \u2192 x \u21a6 x+1"},"headline":"A033282 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a27d9d8a0e13bbdd","kind":"fingerprint_related","node_a_id":"n-c0c21fffce187dd8","node_b_id":"n-2bcf501d4853538e","fingerprint":"f6-ca6d936c14642db2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ca6d936c14642db2"},"other_id":"n-c0c21fffce187dd8","other":{"id":"n-c0c21fffce187dd8","name":"mobius_x_over_x_plus_one of basis_map_A007318 of narayana","first_index":1,"fingerprint":"f6-ca6d936c14642db2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["5","15","11"]],"card_rows":[["1"],["2","3"],["5","15","11"],["14","63","93","45"]],"row_sums_preview":["1","5","31","215","1597","12425","99955","824675"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-58b8cdf8af6ac600","e-127466690b9c3cb0"],"text":"narayana \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"narayana \u2192 basis A007318 \u2192 x \u21a6 x/(x+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:      4,       6;\n  2:     15,      45,       33;\n  3:     56,     252,      372,      180;\n  4:    210,    1260,     2800,     2730,       985;\n  5:    792,    5940,    17640,    25920,     18840,      5418;\n  6:   3003,   27027,   100485,   197505,    216405,    125307,     29953;\n  7:  11440,  120120,   536536,  1321320,   1937320,   1690920,    813400,   166344;\n  8:  43758,  525096,  2738736,  8108100,  14900886,  17405388,  12618144,  5190372,   927441;\n  9: 167960, 2267460, 13525200, 46781280, 103390560, 151400340, 146884920, 91035360, 32704380, 5188590;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n+(2*n^2+4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (4*n^2+6*n+2+(17*n^2+21*n+4)*x+(26*n^2+24*n-2)*x^2+(17*n^2+9*n-8)*x^3+(4*n^2-4)*x^4)*P(n-1,x) - ((2*n+2)*x+(10*n+10)*x^2+(18*n+18)*x^3+(14*n+14)*x^4+(4*n+4)*x^5)*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[((4*n^2 + 6*n + 2 + (17*n^2 + 21*n + 4)*x + (26*n^2 + 24*n - 2)*x^2 + (17*n^2 + 9*n - 8)*x^3 + (4*n^2 - 4)*x^4)*P[n - 1, x] - ((2*n + 2)*x + (10*n + 10)*x^2 + (18*n + 18)*x^3 + (14*n + 14)*x^4 + (4*n + 4)*x^5)*D[P[n - 1, x], x])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^2)]];\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] = ((4*n^2 + 6*n + 2 + (17*n^2 + 21*n + 4)*x + (26*n^2 + 24*n - 2)*x^2 + (17*n^2 + 9*n - 8)*x^3 + (4*n^2 - 4)*x^4)*v[i - 1] - ((2*n + 2)*x + (10*n + 10)*x^2 + (18*n + 18)*x^3 + (14*n + 14)*x^4 + (4*n + 4)*x^5)*deriv(v[i - 1], x))/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^2)); 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-984320118b653f6b","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-2345ffa617ccabd7","e-6d171e78219b20a8"],"text":"A132812 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A132812 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}
