{"id":"n-2790b5ebdbaa03f1","revision":1,"first_index":0,"fingerprint":"f6-1862d174d11ab180","name":"derivative of Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= 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":true,"rows":[[],["4"],["16","20"],["40","100","60"],["80","300","360","140"],["140","700","1260","980","280"],["224","1400","3360","3920","2240","504"],["336","2520","7560","11760","10080","4536","840"],["480","4200","15120","29400","33600","22680","8400","1320"],["660","6600","27720","64680","92400","83160","46200","14520","1980"],["880","9900","47520","129360","221760","249480","184800","87120","23760","2860"],["1144","14300","77220","240240","480480","648648","600600","377520","154440","37180","4004"],["1456","20020","120120","420420","960960","1513512","1681680","1321320","720720","260260","56056","5460"],["1820","27300","180180","700700","1801800","3243240","4204200","3963960","2702700","1301300","420420","81900","7280"],["2240","36400","262080","1121120","3203200","6486480","9609600","10570560","8648640","5205200","2242240","655200","116480","9520"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","36","200","880","3360","11648","37632","115200","337920","957440","2635776","7081984","18636800","48168960"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-4","-2"],["4","1"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["8","14","6"],["4","9","5"],["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164205,"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-11ed69f9c083a573"]},{"id":164206,"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-11ed69f9c083a573"]},{"id":164207,"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-11ed69f9c083a573"]}],"region_exact":{},"incoming_edges":[{"id":"e-11ed69f9c083a573","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-2790b5ebdbaa03f1","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":"030e79b065dcb6e8093762caf209f8020b8eeb1c67e5d1f8e1a274d024e4167d","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-32779d2065847504","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.","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":[],"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:   4;\n  2:  16,   20;\n  3:  40,  100,    60;\n  4:  80,  300,   360,   140;\n  5: 140,  700,  1260,   980,   280;\n  6: 224, 1400,  3360,  3920,  2240,   504;\n  7: 336, 2520,  7560, 11760, 10080,  4536,   840;\n  8: 480, 4200, 15120, 29400, 33600, 22680,  8400,  1320;\n  9: 660, 6600, 27720, 64680, 92400, 83160, 46200, 14520, 1980;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4+(n^2+n-2)*x)*P(n,x) = (4*n+8+(n^2+9*n+14)*x+(n^2+5*n+6)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + 8 + (n^2 + 9*n + 14)*x + (n^2 + 5*n + 6)*x^2)*P[n - 1, x])/(4*n - 4 + (n^2 + n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 8 + (n^2 + 9*n + 14)*x + (n^2 + 5*n + 6)*x^2)*v[i - 1])/(4*n - 4 + (n^2 + n - 2)*x)); 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-c8c19eba9d68881f","steps":["(1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2","D"],"edges":["e-6ba022abf95075ed","e-11ed69f9c083a573"],"text":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 D"},"headline":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 D","chain":null}