{"id":"n-4efe71505c1f5edb","revision":0,"first_index":1,"fingerprint":"f6-6832a290ef8b97c2","name":"derivative of derivative of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["2"],["24","6"],["180","120","12"],["1120","1260","360","20"],["6300","10080","5040","840","30"],["33264","69300","50400","15120","1680","42"],["168168","432432","415800","184800","37800","3024","56"],["823680","2522520","3027024","1801800","554400","83160","5040","72"],["3938220","14002560","20180160","15135120","6306300","1441440","166320","7920","90"],["18475600","74826180","126023040","114354240","60540480","18918900","3363360","308880","11880","110"],["85357272","387987600","748261800","798145920","514594080","205837632","50450400","7207200","540540","17160","132"],["389398464","1963217256","4267863600","5237832600","3990729600","1955457504","617512896","122522400","14414400","900900","24024","156"],["1757701400","9734961600","23558607072","32720287600","28808079300","16761064320","6518191680","1676106432","275675400","27227200","1441440","32760","182"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","2","30","312","2760","22290","169806","1242080","8817696","61178130","416822670","2798399736","18559873800","121839376386"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["9"],["-6"],["1"]]},"terms":[{"coeff":{"coeffs":[["20","19"],["-18","-16"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","4"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178466,"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-4fff5accea5d65f2"]},{"id":178467,"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-4fff5accea5d65f2"]},{"id":178468,"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-4fff5accea5d65f2"]}],"region_exact":{},"incoming_edges":[{"id":"e-4fff5accea5d65f2","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-4efe71505c1f5edb","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":"92ea809192fcf2fcfd9d87f1833f05735a3e09ba808afa9a92b267cdd5c64018","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-1ff7b1d8c09df87e","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","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-631cf37755c5a09f","kind":"fingerprint_related","node_a_id":"n-fc24cdc1c25a5a29","node_b_id":"n-4efe71505c1f5edb","fingerprint":"f6-6832a290ef8b97c2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6832a290ef8b97c2"},"other_id":"n-fc24cdc1c25a5a29","other":{"id":"n-fc24cdc1c25a5a29","name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-6832a290ef8b97c2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","10","15"]],"card_rows":[["1"],["1","4"],["1","10","15"],["1","18","63","56"]],"row_sums_preview":["1","5","26","138","743","4043","22180","122468"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ca874876022e136","steps":["x \u21a6 x/(x+1)"],"edges":["e-d56f4dbb61367368"],"text":"A062196 \u2192 x \u21a6 x/(x+1)"},"headline":"A062196 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7a0fe489936d37ff","kind":"fingerprint_related","node_a_id":"n-4efe71505c1f5edb","node_b_id":"n-1ea5485793a82d6a","fingerprint":"f6-6832a290ef8b97c2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6832a290ef8b97c2"},"other_id":"n-1ea5485793a82d6a","other":{"id":"n-1ea5485793a82d6a","name":"derivative of Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.","first_index":0,"fingerprint":"f6-6832a290ef8b97c2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["6","24"]],"card_rows":[["2"],["6","24"],["24","240","360"],["120","2160","7560","6720"]],"row_sums_preview":["0","2","30","624","16560","534960","20376720","894297600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4cba07695b3e3f90","steps":["D"],"edges":["e-eac6eb6e31ecc5f3"],"text":"A357367 \u2192 D"},"headline":"A357367 \u2192 D","chain":null}},{"id":"l-b6f7e97cd4d0eae5","kind":"fingerprint_related","node_a_id":"n-09c4d3901836573e","node_b_id":"n-4efe71505c1f5edb","fingerprint":"f6-6832a290ef8b97c2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6832a290ef8b97c2"},"other_id":"n-09c4d3901836573e","other":{"id":"n-09c4d3901836573e","name":"shift_x_plus_one of reversal of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-6832a290ef8b97c2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["15","10","1"]],"card_rows":[["1"],["4","1"],["15","10","1"],["56","63","18","1"]],"row_sums_preview":["1","5","26","138","743","4043","22180","122468"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-66e2c3448245b0b7","steps":["\u03b8+1","reverse"],"edges":["e-10c4942a23949b48","e-263d479d9ce33dde"],"text":"A033282 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A033282 \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-e4a147494317a8b9","kind":"fingerprint_related","node_a_id":"n-9d7951b6c7c263fc","node_b_id":"n-4efe71505c1f5edb","fingerprint":"f6-6832a290ef8b97c2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6832a290ef8b97c2"},"other_id":"n-9d7951b6c7c263fc","other":{"id":"n-9d7951b6c7c263fc","name":"derivative of Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).","first_index":0,"fingerprint":"f6-6832a290ef8b97c2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["3","12"]],"card_rows":[["2"],["3","12"],["4","40","60"],["5","90","315","280"]],"row_sums_preview":["0","2","15","104","690","4458","28301","177440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2fa133578d3075fe","steps":["D"],"edges":["e-b5b99f0178c1805c"],"text":"A386789 \u2192 D"},"headline":"A386789 \u2192 D","chain":null}},{"id":"l-e9e933c1cb304ab7","kind":"fingerprint_related","node_a_id":"n-7b4f74988046113d","node_b_id":"n-4efe71505c1f5edb","fingerprint":"f6-6832a290ef8b97c2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6832a290ef8b97c2"},"other_id":"n-7b4f74988046113d","other":{"id":"n-7b4f74988046113d","name":"theta_plus_one 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":"f6-6832a290ef8b97c2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","10","15"]],"card_rows":[["1"],["1","4"],["1","10","15"],["1","18","63","56"]],"row_sums_preview":["1","5","26","138","743","4043","22180","122468"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-66e2c3448245b0b7","steps":["\u03b8+1"],"edges":["e-10c4942a23949b48"],"text":"A033282 \u2192 \u03b8+1"},"headline":"A033282 \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:      0;\n   2:      0;\n   3:      2;\n   4:     24,       6;\n   5:    180,     120,      12;\n   6:   1120,    1260,     360,      20;\n   7:   6300,   10080,    5040,     840,     30;\n   8:  33264,   69300,   50400,   15120,   1680,    42;\n   9: 168168,  432432,  415800,  184800,  37800,  3024,   56;\n  10: 823680, 2522520, 3027024, 1801800, 554400, 83160, 5040, 72;\n","recurrence_sentence":"The row polynomials satisfy (n^2-6*n+9)*P(n,x) = (4*n^2-18*n+20+(3*n^2-16*n+19)*x)*P(n-1,x) - ((2*n-4)*x+(2*n-4)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 18*n + 20 + (3*n^2 - 16*n + 19)*x)*P[n - 1, x] - ((2*n - 4)*x + (2*n - 4)*x^2)*D[P[n - 1, x], x])/(n^2 - 6*n + 9)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = ((4*n^2 - 18*n + 20 + (3*n^2 - 16*n + 19)*x)*v[i - 1] - ((2*n - 4)*x + (2*n - 4)*x^2)*deriv(v[i - 1], x))/(n^2 - 6*n + 9)); 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-78ba05b4a3dd925d","steps":["D","D"],"edges":["e-4c8816dc3f762029","e-4fff5accea5d65f2"],"text":"A104684 \u2192 D \u2192 D"},"headline":"A104684 \u2192 D \u2192 D","chain":null}
