{"id":"n-c5e1848c5b60987e","revision":0,"first_index":0,"fingerprint":"f6-f1789e1ed3fa2160","name":"derivative of reversal of Triangle of Dyck paths counted by number of long interior inclines.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[[],["3"],["8","8"],["15","40","15"],["24","120","120","24"],["35","280","525","280","35"],["48","560","1680","1680","560","48"],["63","1008","4410","7056","4410","1008","63"],["80","1680","10080","23520","23520","10080","1680","80"],["99","2640","20790","66528","97020","66528","20790","2640","99"],["120","3960","39600","166320","332640","332640","166320","39600","3960","120"],["143","5720","70785","377520","990990","1359072","990990","377520","70785","5720","143"],["168","8008","120120","792792","2642640","4756752","4756752","2642640","792792","120120","8008","168"],["195","10920","195195","1561560","6441435","14723280","19324305","14723280","6441435","1561560","195195","10920","195"],["224","14560","305760","2914912","14574560","41225184","68708640","68708640","41225184","14574560","2914912","305760","14560","224"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","16","70","288","1155","4576","18018","70720","277134","1085280","4249388","16640960","65189475","255487680"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"denominator":{"coeffs":[["-3"],["-1"],["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["6","-2"],["5","5"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","4","-4"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174634,"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-df22964a4691c109"]},{"id":174635,"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-df22964a4691c109"]},{"id":174636,"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-df22964a4691c109"]},{"id":174637,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-df22964a4691c109"]}],"region_exact":{},"incoming_edges":[{"id":"e-df22964a4691c109","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-c5e1848c5b60987e","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d60556eb3eb205190f6c04ae06f740f412f7ad6b1627307ba6479574ee853c6e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c7016e67374634ec","role":"sequence","name":"reversal of Triangle of Dyck paths counted by number of long interior inclines.","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-dd754c66b5288c4c","kind":"fingerprint_related","node_a_id":"n-c5e1848c5b60987e","node_b_id":"n-18e89b8a7a384861","fingerprint":"f6-f1789e1ed3fa2160","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f1789e1ed3fa2160"},"other_id":"n-18e89b8a7a384861","other":{"id":"n-18e89b8a7a384861","name":"derivative of derivative of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-f1789e1ed3fa2160","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["12"]],"card_rows":[["12"],["60","60"],["180","480","180"],["420","2100","2100","420"]],"row_sums_preview":["0","0","12","120","840","5040","27720","144144"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ca874876022e136","steps":["D","D"],"edges":["e-7729d1773ce70fbd","e-4d34f3eec5ee1c70"],"text":"A062196 \u2192 D \u2192 D"},"headline":"A062196 \u2192 D \u2192 D","chain":null}},{"id":"l-e8c312f0cbb87be1","kind":"fingerprint_related","node_a_id":"n-0e6d847331141ca6","node_b_id":"n-c5e1848c5b60987e","fingerprint":"f6-f1789e1ed3fa2160","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f1789e1ed3fa2160"},"other_id":"n-0e6d847331141ca6","other":{"id":"n-0e6d847331141ca6","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","first_index":0,"fingerprint":"f6-f1789e1ed3fa2160","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["3","8","3"]],"card_rows":[["1"],["2","2"],["3","8","3"],["4","20","20","4"]],"row_sums_preview":["1","4","14","48","165","572","2002","7072"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, +inf)"],"region_exact":{},"oeis_ids":["A145596"],"derivation":{"root":"n-0e6d847331141ca6","steps":[],"edges":[],"text":"A145596"},"headline":"A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by 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:  0;\n  1:  3;\n  2:  8,    8;\n  3: 15,   40,    15;\n  4: 24,  120,   120,    24;\n  5: 35,  280,   525,   280,    35;\n  6: 48,  560,  1680,  1680,   560,    48;\n  7: 63, 1008,  4410,  7056,  4410,  1008,    63;\n  8: 80, 1680, 10080, 23520, 23520, 10080,  1680,   80;\n  9: 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2-n-3)*P(n,x) = (n^3+5*n^2+6*n+(3*n^3+5*n^2-2*n)*x)*P(n-1,x) + ((2*n^2+4*n)*x-(2*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 5*n^2 + 6*n + (3*n^3 + 5*n^2 - 2*n)*x)*P[n - 1, x] + ((2*n^2 + 4*n)*x - (2*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^3 + 3*n^2 - n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 5*n^2 + 6*n + (3*n^3 + 5*n^2 - 2*n)*x)*v[i - 1] + ((2*n^2 + 4*n)*x - (2*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^3 + 3*n^2 - 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-c963abfc0266d66c","steps":["reverse","D"],"edges":["e-7ac651410315272d","e-df22964a4691c109"],"text":"A108838 \u2192 reverse \u2192 D"},"headline":"A108838 \u2192 reverse \u2192 D","chain":null}
