{"id":"n-eced3c738adffaba","revision":0,"first_index":0,"fingerprint":"f6-5f98bd4aea48ed7d","name":"derivative of derivative of Square the entries of Pascal's triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["2"],["18","6"],["72","96","12"],["200","600","300","20"],["450","2400","2700","720","30"],["882","7350","14700","8820","1470","42"],["1568","18816","58800","62720","23520","2688","56"],["2592","42336","190512","317520","211680","54432","4536","72"],["4050","86400","529200","1270080","1323000","604800","113400","7200","90"],["6050","163350","1306800","4268880","6403320","4573800","1524600","217800","10890","110"],["8712","290400","2940300","12545280","25613280","26345088","13721400","3484800","392040","15840","132"],["12168","490776","6134700","33127380","88339680","123675552","92756664","36808200","7361640","669240","22308","156"],["16562","794976","12024012","80160080","270540270","494702208","505008504","288576288","90180090","14574560","1093092","30576","182"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","2","24","180","1120","6300","33264","168168","823680","3938220","18475600","85357272","389398464","1757701400"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["4"],["-4"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","6"],["0","-10"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","2"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176015,"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-55c691d134f6467c"]},{"id":176016,"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-55c691d134f6467c"]},{"id":176017,"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-55c691d134f6467c"]}],"region_exact":{},"incoming_edges":[{"id":"e-55c691d134f6467c","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-eced3c738adffaba","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":"63ea92b9d6fbe9e316f084676058fa905f75066eb42bb7fb8e815a9fb381a6f0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-aff68976ca55ced5","role":"sequence","name":"derivative of Square the entries of Pascal's triangle.","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-02847a9cf651bbe3","kind":"fingerprint_related","node_a_id":"n-b229ca007af8822e","node_b_id":"n-eced3c738adffaba","fingerprint":"f6-5f98bd4aea48ed7d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5f98bd4aea48ed7d"},"other_id":"n-b229ca007af8822e","other":{"id":"n-b229ca007af8822e","name":"derivative of Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.","first_index":0,"fingerprint":"f6-5f98bd4aea48ed7d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["2","6"]],"card_rows":[["2"],["2","6"],["2","16","12"],["2","30","60","20"]],"row_sums_preview":["0","2","8","30","112","420","1584","6006"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8a65e165240c40e0","steps":["D"],"edges":["e-c15b58e3faa995cd"],"text":"A281260 \u2192 D"},"headline":"A281260 \u2192 D","chain":null}},{"id":"l-21e891093e5a3514","kind":"fingerprint_related","node_a_id":"n-4ca874876022e136","node_b_id":"n-eced3c738adffaba","fingerprint":"f6-5f98bd4aea48ed7d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5f98bd4aea48ed7d"},"other_id":"n-4ca874876022e136","other":{"id":"n-4ca874876022e136","name":"Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-5f98bd4aea48ed7d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","8","6"]],"card_rows":[["1"],["1","3"],["1","8","6"],["1","15","30","10"]],"row_sums_preview":["1","4","15","56","210","792","3003","11440"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A062196"],"derivation":{"root":"n-4ca874876022e136","steps":[],"edges":[],"text":"A062196"},"headline":"A062196: Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","chain":null}},{"id":"l-38b1b57c2688e8bc","kind":"fingerprint_related","node_a_id":"n-eced3c738adffaba","node_b_id":"n-5b2cf56653b8394b","fingerprint":"f6-5f98bd4aea48ed7d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5f98bd4aea48ed7d"},"other_id":"n-5b2cf56653b8394b","other":{"id":"n-5b2cf56653b8394b","name":"theta_plus_one of derivative of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-5f98bd4aea48ed7d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["8","24"]],"card_rows":[["3"],["8","24"],["15","120","90"],["24","360","720","240"]],"row_sums_preview":["0","3","32","225","1344","7350","38016","189189"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ca874876022e136","steps":["D","\u03b8+1"],"edges":["e-7729d1773ce70fbd","e-5fed6b956a080c72"],"text":"A062196 \u2192 D \u2192 \u03b8+1"},"headline":"A062196 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-4add6518b3c416c7","kind":"fingerprint_related","node_a_id":"n-0d8e75150d012303","node_b_id":"n-eced3c738adffaba","fingerprint":"f6-5f98bd4aea48ed7d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5f98bd4aea48ed7d"},"other_id":"n-0d8e75150d012303","other":{"id":"n-0d8e75150d012303","name":"theta_plus_one of reversal of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-5f98bd4aea48ed7d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2","6"],["2","16","12"]],"card_rows":[["2"],["2","6"],["2","16","12"],["2","30","60","20"]],"row_sums_preview":["2","8","30","112","420","1584","6006","22880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["reverse","\u03b8+1"],"edges":["e-7ac651410315272d","e-8d225a11e66b4eb8"],"text":"A108838 \u2192 reverse \u2192 \u03b8+1"},"headline":"A108838 \u2192 reverse \u2192 \u03b8+1","chain":null}},{"id":"l-5a0f4f6e5e59aa76","kind":"fingerprint_related","node_a_id":"n-3f65f4dc3d5d123d","node_b_id":"n-eced3c738adffaba","fingerprint":"f6-5f98bd4aea48ed7d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5f98bd4aea48ed7d"},"other_id":"n-3f65f4dc3d5d123d","other":{"id":"n-3f65f4dc3d5d123d","name":"reversal of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-5f98bd4aea48ed7d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["6","8","1"]],"card_rows":[["1"],["3","1"],["6","8","1"],["10","30","15","1"]],"row_sums_preview":["1","4","15","56","210","792","3003","11440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ca874876022e136","steps":["reverse"],"edges":["e-3f411cadac720071"],"text":"A062196 \u2192 reverse"},"headline":"A062196 \u2192 reverse","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:    0;\n  2:    2;\n  3:   18,     6;\n  4:   72,    96,     12;\n  5:  200,   600,    300,     20;\n  6:  450,  2400,   2700,    720,     30;\n  7:  882,  7350,  14700,   8820,   1470,    42;\n  8: 1568, 18816,  58800,  62720,  23520,  2688,   56;\n  9: 2592, 42336, 190512, 317520, 211680, 54432, 4536, 72;\n","recurrence_sentence":"The row polynomials satisfy (n^2-4*n+4)*P(n,x) = (n^2+(3*n^2-10*n+6)*x)*P(n-1,x) + ((2*n-2)*x-(2*n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + (3*n^2 - 10*n + 6)*x)*P[n - 1, x] + ((2*n - 2)*x - (2*n - 2)*x^2)*D[P[n - 1, x], x])/(n^2 - 4*n + 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 + (3*n^2 - 10*n + 6)*x)*v[i - 1] + ((2*n - 2)*x - (2*n - 2)*x^2)*deriv(v[i - 1], x))/(n^2 - 4*n + 4)); 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-b13531ad79e7c629","steps":["D","D"],"edges":["e-be8e3a0f5d3685bc","e-55c691d134f6467c"],"text":"A008459 \u2192 D \u2192 D"},"headline":"A008459 \u2192 D \u2192 D","chain":null}